Asymptotics for Fermi curves of electric and magnetic periodic fields by Gustavo de Oliveira A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) July 2009 c©Gustavo de Oliveira, 2009 Abstract This work is concerned with some geometrical properties of (complex) Fermi curves of electric and magnetic periodic fields. These are analytic curves in C2 that arise from the study of the eigenvalue problem for periodic Schrödinger operators. More specifically, we characterize a certain class of these curves in the region of C2 where at least one of the coordinates has “large” imaginary part. The new results obtained in this thesis extend previous results in the absence of magnetic field to the case of “small” magnetic field. Our theorems can be used to show that generically these Fermi curves belong to a class of Riemann surfaces of infinite genus. ii Table of contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi 1 Introduction and summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Periodic Schrödinger operators . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Bloch theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Fermi surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Electrons in a crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Basic properties of Hk(A, V ) . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 The complex analytic structure of the spectrum . . . . . . . . . . . . . . . . 26 2.6 Gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3 Asymptotics for Fermi curves . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1 Fermi curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 The free Fermi curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 The ε-tubes about the free Fermi curve . . . . . . . . . . . . . . . . . . . . 41 3.4 Motivation and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.5 Strategy outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.6 Notation and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4 Weak magnetic potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.1 Invertibility of RG′G′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 iii 4.2 Local defining equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.3 Change of coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.4 Asymptotics for the coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.5 Bounds on the derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.6 The regular piece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.7 The handles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.8 Quantitative Morse lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5 Exploiting gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.1 A gauge transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.2 The regular piece revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 iv List of figures 2.1 Sketch of band functions En(k). . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Construction of D(m) ⊂ R3: gluing together an m×m×m set of cells D ⊂ R3. 13 2.3 Energy bands in conductors and insulators. . . . . . . . . . . . . . . . . . . 14 3.1 Sketch of F̂(0, 0) and F(0, 0) when both ik1 and k2 are real. . . . . . . . . . 39 3.2 The pair of helices as a “manifold”. . . . . . . . . . . . . . . . . . . . . . . . 40 3.3 The full F̂(0, 0): two copies of C with θ1(b) and θ2(b) identified for b2 6= 0. . 40 3.4 On the left: definition of Λ. On the right: possible configuration. . . . . . . 42 3.5 The ε-tubes about the free “lifted” Fermi curve. . . . . . . . . . . . . . . . 44 3.6 Sketch of (∪d′∈GTd′) ∩ (C2 \ ∪b∈G′Tb) for G = {0} and G = {0, d}. . . . . . . 52 4.1 Sketch of ( T0 \ ∪b∈Γ#\{0}Tb ) \ KR (left) and (T0 ∩ Td) \ KR (right). . . . . . 60 v Acknowledgements I would like to thank Professor Joel Feldman for suggesting this research topic, and for his help and support during the process of writing this thesis. I am also thankful to UBC and all its staff. This thesis was typeset using LATEX and the many packages that accompany the LATEX distribution. In particular, the graphics were created using the TikZ and pgf packages. All the work was done using a free and open source operating system. My thanks go to the members of the “free software community”, who have made available such powerful tools. vi Chapter 1 Introduction and summary This work is concerned with some geometrical properties of (complex) Fermi curves. These are analytic curves in C2 that arise from the study of the eigenvalue problem for periodic Schrödinger operators. More specifically, we characterize a certain class of these curves in the region of C2 where at least one of the coordinates has “large” imaginary part. In order to describe the contents of this thesis we need to introduce the definition of Fermi curve. We stress that our purpose here is only to grasp the main ideas. We will return to the definitions (and results) later in much more detail. Let A1, A2 and V be functions on R2 that are periodic with respect to Z2 and consider the operator H = (i∇+A)2 + V acting on L2(R2), where ∇ is the gradient operator in R2 and A = (A1, A2). For k ∈ C2 let En(k) be the n-th eigenvalue of the boundary value problem Hψ = λψ, ψ(x+ γ) = eik·γψ(x) for all x ∈ R2 and all γ ∈ Z2. By definition, the n-th band Bn ⊂ C is the range of the function k 7→ En(k). It is known that the spectrum of H is the union of the bands Bn (restricted to real k) for n ≥ 1. The (complex) Fermi curve of A and V with energy λ ∈ C is defined as Fλ,A,V = {k ∈ C2 | the above boundary value problem has a nonzero solution ψ}. Similarly one can define Fermi “surfaces” in any dimension greater than two. 1 The above operator H (and its three-dimensional counterpart) is important in solid state physics. It is the Hamiltonian of a single electron under the influence of the magnetic field with vector potential A, and the electric field with scalar potential V , in the independent electron model of a two-dimensional solid [12]. The classical framework for studying the spectrum of a differential operator with periodic coefficients is the Floquet (or Bloch) theory [12, 8]. Roughly speaking, the main idea of this theory is to “decompose” the original eigenvalue problem, which usually has continuous spectrum, into a family of boundary value problems, each one having discrete spectrum. In our context this leads to decomposing the problem Hψ = λψ (without boundary conditions) into the above k-family of boundary value problems. The Fermi surface—and in particular the Fermi curve—has the following remarkable property. There exists a function F (k, λ,A, V ) analytic on Cd×C×A×V, where A and V are suitable spaces of functions, such that Fλ,A,V = {k ∈ Cd | F (k, λ,A, V ) = 0}. In other words, the Fermi surface is a complex analytic variety. In Chapter 2 we provide a detailed proof of this (well-known) theorem following the proof in [3]. It is believed that for “generic” sufficiently regular potentials A and V the union of the surfaces z = En(k) for n ≥ 1 is a complex analytic manifold. In fact, this statement was made precise in [4] for Fermi curves without magnetic potential (A = 0), and for heat curves. The later are the spectral curves (defined similarly as the Fermi curves) associated to the “heat” equation ψx1 − ψx2x2 + V ψ = 0 with V periodic, while the former are a particular case of the curves studied in this thesis (where A 6= 0). This picture is believed to hold for other differential operators with sufficiently regular periodic coefficients as well. In Chapters 3 and 4 we prove a theorem, that holds only for “small” A, which is the main step for showing that the above picture is also true for Fermi curves with “small” magnetic potential. This is the main contribution of this thesis. We have followed the same strategy that Feldman, Knörrer and Trubowitz implemented in [4, §16] for proving a similar result for A = 0. Below we provide more details about our results. As we have already mentioned in the last paragraph, there is a relationship between Riemann surfaces and differential operators with periodic coefficients. We briefly mention two examples here. In one dimension, the solution of the KdV equation ut = 3uux − 12uxxx with initial data u0 is related to the Schrödinger curve S(V ) by S(u( · , t)) = S(u0), where 2 S(V ) is the spectral curve associated to the one-dimensional analogue of the operator H with A = 0 [10]. In two dimensions, there is a relationship between Riemann surfaces of finite genus and solutions of the KP equation ux1x1 + 2 3(ut + 3uux2 + 1 2ux2x2x2)x2 = 0 [8]. If the initial data for this problem is a function on R2 that is periodic with respect to a certain lattice, this relationship is even more explicit [4]. In this case the solution of the KP equation is related to the heat curve mentioned above. This turns out to be, for “generic” potentials V , a Riemann surface of infinite genus according to the theory proposed in [4]. The class of surfaces introduced in that work yields an extension of the classical theory of finite genus that has analogues of many theorems of finite genus theory. When A and V are zero the (free) Fermi curve can be found explicitly. It consists of two copies of C with the points −b2 + ib1 (in the first copy) and b2 + ib1 (in the second copy) identified for all (b1, b2) ∈ piZ2 with b2 6= 0. The purpose of this thesis is to show that in the region of C2 where k ∈ C2 has “large” imaginary part the Fermi curve (for nonzero A and V ) is “close” to the free Fermi curve. When A is zero this was proved by Feldman, Knor̈rer and Trubowitz in [4, §16]. Very briefly, the main result of this thesis is essentially the following. “Theorem”. Suppose that A and V have some regularity and assume that A is sufficiently small in a suitable norm. Write k in C2 as k = u + iv with u and v in R2 and suppose that |v| is sufficiently large. (Recall that the free Fermi curve is two copies of C with certain points in one copy identified with points in the other one.) Then, in this region of C2, the Fermi curve of A and V is very close to the free Fermi curve, except that instead of two planes we may have two deformed planes, and identifications between points can open up to handles that look like {(z1, z2) ∈ C2 | z1z2 = constant} in suitable local coordinates. To prove this theorem we follow the same strategy as [4]. The proof has basically three steps. We first derive very detailed information about the free Fermi curve (which is explicitly known). This is done in Chapter 3. Then, to compute the interacting Fermi curve we have to find the kernel of H − λI in L2(R2) with the above boundary condition. In the second step of the proof we derive a number of estimates for showing that this kernel has finite dimension for small A and k ∈ C2 with large imaginary part. Furthermore, in the complement of this kernel in L2(R2), after a suitable (invertible) change of variables in L2(R2), the operator H −λI multiplied by the inverse of the operator that implements this 3 change of variables is a (compact) perturbation of the identity. This reduces the problem of finding the kernel to finite dimension and thus we can write (local) defining equations for the Fermi curve. In the third step of the proof we use these equations to study the Fermi curve. A few more estimates and the implicit function theorem gives us the deformed planes. The handles are obtained using a quantitative Morse lemma available in [4] that we prove at end of Chapter 4. Steps two and three are contained in Chapter 4, which is the core of this thesis. We have not yet mentioned an important property of H—namely, gauge invariance. Briefly, gauge invariance implies that the spectrum of (i∇ + A)2 + V is the same as the spectrum of (i∇+ A+∇Ψ)2 + V , where Ψ is function on R2 (under suitable hypotheses), and ∇Ψ is periodic with respect to Z2. Consequently, the Fermi curve of A and V is equal to the Fermi curve of A + ∇Ψ and V . In Chapter 5, by choosing a convenient gauge Ψ, we are able to indicate how to simplify the proof of the above theorem and improve some constants in it. After performing this gauge transformation “some terms vanish” and the analysis becomes simpler. The critical part of the proof is the second step. The main difficulty arises due to the presence of the term A · i∇ in H. When A is large, taking the imaginary part of k ∈ C2 arbitrarily large is not enough to control this term—it is not enough to make its contribution small and hence have the interacting Fermi curve as a perturbation of the free Fermi curve. (The term V in H is easily controlled by this method.) However, the proof can be implemented by assuming that A is small. We finally make some remarks about the interdependence of chapters. Since we have attempted to write this thesis in order to minimize referencing, the reader may notice some minor repetitions along the text. Chapter 2 is completely independent of the others; Chapter 3 is independent and only needs Chapter 4 to be fulfilled; Chapter 4 depends on Chapter 3, and Chapter 5 depends on Chapters 3 and 4, and can be seen as a bonus chapter. The most demanding (but still elementary) part to read in this thesis is the proof of the lemmas and propositions in Chapter 4. These proofs may be (safely) skipped in a first reading. 4 Chapter 2 Periodic Schrödinger operators 2.1 Bloch theory Below we briefly describe the main ideas of Bloch (or Floquet) theory without providing proofs. Our discussion is based on [2, 12]. Let Γ be a lattice of static ions in Rd and suppose that the ions generate an electric potential V (x) and a magnetic potential A(x) = (A1(x), . . . , Ad(x)) that are periodic with respect to Γ. Then the Hamiltonian of a single electron moving in Rd under the influence of this lattice is H = (i∇+A)2 + V, where ∇ is the gradient operator in Rd. This operator acts on L2(Rd). For simplicity we suppress the physical constants in the Hamiltonian. For γ ∈ Γ consider the translation operator Tγ acting on L2(Rd) as Tγ : ϕ(x) 7−→ ϕ(x+ γ). This operator is unitary on L2(Rd) and TγTγ′ = Tγ+γ′ for all γ, γ′ ∈ Γ. Furthermore, it is not difficult to verify that the Hamiltonian H (formally) commutes with all the translation operators, HTγ = TγH for all γ ∈ Γ. Under suitable hypotheses on the potentials A and V one can in fact prove that this property holds on an appropriate domain. This translational symmetry is the main ingredient of Bloch theory. 5 Since our goal here is only describing the main ideas of Bloch theory without providing proofs, for the rest of this section we pretend that H and Tγ are matrices. A rigorous version of the procedure below can be implemented for the actual operators. The matrices {H and Tγ for all γ ∈ Γ} form a family of commuting normal matrices. Thus, there exists an orthonormal basis of simultaneous eigenvectors {ϕα} for this family. For all γ ∈ Γ these eigenvectors obey Hϕα = Eαϕα, Tγϕα = λα,γϕα, where Eα and λα,γ are numbers. Let {v1, . . . , vd} be a basis for Γ. Then for any γ ∈ Γ there are integers n1, . . . , nd such that γ = n1v1 + · · ·+ ndvd. If we set γi = nivi for 1 ≤ i ≤ d, then we can write any element γ ∈ Γ as γ = γ1 + · · ·+ γd. As the operator Tγ is unitary, all its eigenvalues are complex numbers of modulus one. Hence, there exist real numbers βα,γ such that λα,γ = eiβα,γ . Using the properties of Tγ we find that eiβα,γ+γ′ϕα = Tγ+γ′ ϕα = TγTγ′ ϕα = Tγ eiβα,γ′ϕα = eiβα,γeiβα,γ′ϕα = ei(βα,γ+βα,γ′ )ϕα, so that βα,γ+γ′ = βα,γ + βα,γ′ mod 2pi for all γ, γ′ ∈ Γ. Consequently, βα,γ = βα,γ1+ ···+γd = βα,γ1 + · · ·+ βα,γd mod 2pi for all γ ∈ Γ. That is, the number βα,γ is determined (mod 2pi) by βα,γ1 , . . . , βα,γd . Now observe that, given any d numbers β1, . . . , βd, the system of linear equations γi · k = βi for 1 ≤ i ≤ d for the unknowns k1, . . . , kd has a unique solution. This follows from the fact that γ1, . . . , γd are linear independent. Hence, for each α there exists a kα ∈ Rd such that γi · kα = βα,γi for 1 ≤ i ≤ d. Therefore, βα,γ = γ · kα mod 2pi 6 for all γ ∈ Γ. Note that, for each α the vector kα is not uniquely determined. Indeed, βα,γ = γ · kα mod 2pi and βα,γ = γ · k′α mod 2pi for all γ ∈ Γ, if and only if (kα − k′α) · γ ∈ 2piZ for all γ ∈ Γ. If we define the dual lattice of Γ as Γ# = {b ∈ Rd | b · γ ∈ 2piZ for all γ ∈ Γ}, where b · γ is the usual scalar product on Rd, the last expression can be rewritten as kα − k′α ∈ Γ#. Summarizing, the numbers {βα,γ for γ ∈ Γ} determine kα up to a vector in Γ#. Hence, the vector kα is unique in Rd/Γ#. This establishes a correspondence between α and kα. Now, relabel the eigenvalues and eigenvectors by replacing the index α by the corre- sponding vector k ∈ Rd/Γ# and another index n. The index n is needed because many kα with different values of α can be equal. Under the new labelling the eigenvalue-eigenvector equations become Hϕn,k = En(k)ϕn,k, Tγϕn,k = eik·γϕn,k for all γ ∈ Γ. The eigenvalue of H is denoted En(k) rather than En,k because, while k runs over the continuous set Rd/Γ#, it turns out that n runs over a countable set. Observing the definition of Tγ , for each k ∈ Rd the above equations can be rewritten as Hϕn,k = En(k)ϕn,k, ϕn,k(x+ γ) = eik·γϕn,k(x) for all x ∈ Rd and all γ ∈ Γ. Under suitable hypotheses on the potentials A and V this boundary value problem is self-adjoint. As we have already mentioned, its spectrum is discrete, it consists of a sequence of real eigenvalues E1(k) ≤ E2(k) ≤ · · · ≤ En(k) ≤ · · · 7 For each integer n ≥ 1 the eigenvalue En(k) defines a continuous function of k that is periodic with respect to the dual lattice Γ#. It is customary to refer to k as the crystal momentum and to En(k) as the n-th band function. The corresponding normalized eigenfunctions ϕn,k are called Bloch eigenfunctions. Let Uk be the unitary transformation on L2(Rd) that acts as Uk : ϕ(x) 7−→ eik·xϕ(x). By applying this transformation we can rewrite the above problem and put the boundary conditions into the operator. Indeed, if we define Hk = U−1k H Uk and ψn,k = U −1 k ϕn,k, then the above problem is unitary equivalent to Hkψn,k = En(k)ψn,k, ψn,k(x+ γ) = ψn,k(x) for all x ∈ Rd and all γ ∈ Γ, or, using a more compact notation, Hkψn,k = En(k)ψn,k for ψn,k ∈ L2(Rd/Γ). To see that these problems are equivalent we proceed (formally) as follows. On the one hand, from the original problem and using the above transformation we find that 0 = U−1k 0 = U −1 k (H − En(k))ϕn,k = U−1k (H − En(k))UkU−1k ϕn,k = (U−1k HUk − En(k))U−1k ϕn,k = (Hk − En(k))ψn,k and ψn,k(x+ γ) = (U−1k ϕn,k)(x+ γ) = e −ik·(x+γ)ϕn,k(x+ γ) = e−ik·(x+γ)eik·γϕn,k(x) = e−ik·xϕn,k(x) = (U−1k ϕn,k)(x) = ψn,k(x). On the other hand, by a similar computation (in the reverse order), using the last two equalities and the above transformation we derive the original problem. This (formally) implies unitary equivalence. Furthermore, a simple (formal) calculation shows that Hk = (i∇+A− k)2 + V. 8 In fact, Hkψ = U−1k H Ukψ = e−ik·x [ (i∇+A)2 + V ]eik·xψ = e−ik·x [ (i∇+A) · eik·x(−kψ + i∇ψ +Aψ)]+ V ψ = (−k +A) · (−k + i∇+A)ψ + i∇ · (−k + i∇+A)ψ + V ψ = [ (i∇+A− k)2 + V ]ψ. Of course, the unitary transformation Uk preserves self-adjointness and does not change the spectrum {En(k)}∞n=1. Finally, denote by Nk the set of values of n that appear in the pairs α = (k, n) and define Hk = span{ϕn,k | n ∈ Nk}. Then, formally, and in particular ignoring that k runs over an uncountable set, L2(Rd) = span{ϕn,k | k ∈ Rd/Γ# and n ∈ Nk} = ⊕ k∈Rd/Γ# Hk. Set H̃k = span{ψn,k | n ∈ Nk}. Observe that, as the operator U−1k is unitary on L 2(Rd), the space Hk is unitary equivalent to H̃k, and L2(Rd) is unitary equivalent to ⊕k∈Rd/Γ#H̃k. The restriction of U−1k HUk to H̃k is Hk applied to functions that are periodic with respect to Γ. Therefore, at least formally, to find the spectrum of H on L2(Rd) it suffices to find the spectrum of Hk on L2(Rd/Γ) for all k ∈ Rd/Γ#. Unlike H, the operator Hk has compact resolvent. Thus, the spectrum of Hk is discrete, unlike the spectrum of H that is continuous and is given by σ(H) = {En(k) | n ∈ N and k ∈ Rd/Γ#}. (See Figure 2.1 below.) All these statements can be made precise and rigorously proved. We prove a few of them in §2.4; the proof of some others can be found in [2, 12]. 9 k ∈ Rd E1 E2 E3 E4 σ(H) Figure 2.1: Sketch of band functions En(k). 2.2 Fermi surfaces In this section we define the Fermi surfaces in Rd. We first state more precisely some definitions already given above. Let Γ be a lattice of maximal rank in Rd with d ≥ 2. Consider a real number r > d and set A := {(A1, . . . , Ad) | Aj ∈ LrR(Rd/Γ) for 1 ≤ j ≤ d} and V := Lr/2R (Rd/Γ), where LrR(R d/Γ) is the space of real-valued functions f such that |f |r is integrable on the torus Rd/Γ. For real-valued potentials (A, V ) ∈ A× V and for k ∈ Rd define the operator Hk(A, V ) := (i∇+A− k)2 + V acting on L2(Rd/Γ), where ∇ is the gradient operator in Rd/Γ. Recall the discussion in the last section and consider the eigenvalue-eigenvector problem Hk(A, V )ψn,k = En(k,A, V )ψn,k for ψn,k ∈ L2(Rd/Γ). The real “lifted” Fermi surface of (A, V ) with energy λ ∈ R is defined as F̂λ,R(A, V ) := {k ∈ Rd | En(k,A, V ) = λ for some n ≥ 1}. 10 Equivalently, F̂λ,R(A, V ) = {k ∈ Rd | (Hk(A, V )− λ)ψ = 0 for some ψ ∈ DHk(A,V ) \ {0}}, where DHk(A,V ) ⊂ L2(Rd/Γ) denotes the (dense) domain of Hk(A, V ). The adjective “lifted” indicates that F̂λ,R(A, V ) is a subset of Rd rather than Rd/Γ#. When d = 2 the Fermi surfaces are called Fermi curves, in particular. This is the main case of interest in this work. We next describe the physical context in which the concept of Fermi surface arises. 2.3 Electrons in a crystal The purpose of this section is to motivate the definition of Fermi surfaces (for d ≤ 3) and briefly describe its physical meaning. For simplicity we consider the case d = 3. Our discussion is based on [9, 12]. A full rigorous theoretical description of crystalline solids is unavailable. In fact, no one has given an explanation from first principles of why crystals form. That is, no one has proven that a large number of heavy nuclei with enough electrons to produce neutrality, interacting via Coulomb potential, have a ground state that is approximately a crystal. Nevertheless, we may consider a simplified model for solids in attempt to describe some of the observed phenomena. This is what we discuss next. It is observed experimentally that the nuclei in a solid lie more or less in a regular array. For example, the common crystalline phase of iron has crystal structure given approximately by αe1Z⊕αe2Z⊕ α2 (e1 +e2 +e2)Z ⊂ R3, where α ≈ 2.87Å and {ej}3j=1 is the canonical base of R3. Thus, we postulate in our model that at each site of a lattice Γ in R3 there is a fixed nucleus with a number of core electrons. Furthermore, we assume that the solid is filling all of R3. Hence, if we ignore electron-electron interactions, and suppose that the fixed nuclei generate an electric potential V (x) and a magnetic potential A(x) = (A1(x), . . . , A3(x)) that are periodic with respect to Γ, we have a cloud of valence electrons moving in R3 subjected only to the influence of this lattice. Each one of these electrons has Hamiltonian H = 12me (i~∇ + eA)2 + V , where me and e are the mass and the charge of the electron, respectively, and ~ is the Planck constant. This model is known as the independent electron model of solids. We shall outline how to use this model to describe the notion of density of states and give a qualitative explanation of the difference between metal and insulators. 11 Let B be a fundamental cell in R3 for the dual lattice Γ# and let {En(k)}∞n=1 be the eigenvalues of Hk. Given a set X ⊂ R3, denote by |X| its Lebesgue measure. The (inte- grated) density of states measure ρ is the measure on R defined by ρ(−∞, E] := 2|B| ∞∑ n=1 ∣∣{k ∈ B | En(k) ≤ E}∣∣. Since En(k) → ∞ uniformly in k as n → ∞, the number ρ(−∞, E] is finite. Furthermore, one can show that ρ is absolutely continuous with respect to dE, the Lebesgue measure on R. The Radon-Nikodym derivative dρ/dE is usually called the density of states. The (integrated) density of states is a concept of fundamental importance in condensed matter physics. It measures the “number of quantum states per unit volume” below a given energy. In fact, to understand the definition of ρ(−∞, E] given above, suppose for simplicity that for a given energy E there are exactly N bands En(k) such that En(k) < E for all k ∈ B. Then we have ρ(−∞, E] = 2|B| ∞∑ n=1 ∣∣{k ∈ B | En(k) ≤ E}∣∣ = N∑ n=1 2|B| |B| = 2N. That is, the “number of quantum states per volume |B|” below the energy E is 2N . Here, the factor of 2 comes from the Pauli principle. Since we are considering electrons (fermions), this principle asserts that each quantum state corresponding to the energy level En(k) can have at most two electrons. Thus, in the definition of ρ(−∞, E] above, we include a factor of 2 in the numerator. (We shall explain below how to implement the Pauli principle in our model). To explain the importance of ρ for the study of solids we introduce one more concept. Let D be a fundamental cell in R3 for the lattice Γ and, given an integer m, let D(m) be the set of volume m3 |D| obtained by gluing together an m×m×m set of D’s (see figure 2.2). Consider the operator Hm = 1 2me (i~∇p + eA)2 + V on L2(D(m)), where ∇p is the gradient operator with periodic boundary conditions. Let Ω be a subset of R and let Pm(Ω) be the spectral projection for Hm. Define ρm(−∞, E] := 2 m3 dim Pm(−∞, E]. It is possible to show that ρm → ρ as m → ∞ in the sense that ρm(−∞, E] → ρ(−∞, E] for all E ∈ R [12]. We can now return to our model of solids. 12 Dm m (in R3) Figure 2.2: Construction of D(m) ⊂ R3: gluing together an m×m×m set of cells D ⊂ R3. Suppose that each nucleus in free space is surrounded by l electrons. Then in our model we wish to have l valence electrons per unit cell. While we ignore the interaction between electrons, we cannot ignore the Pauli principle. Again, this principle asserts that the quantum state corresponding to each eigenvalue of H can have at most two electrons. How do we take this into account when H does not have eigenfunctions and when there are infinitely many electrons (in our infinite crystal lattice)? We claim that a reasonable way of taking the Pauli principle into account is to say that in the ground state the electrons fill up the continuum eigenstates up to that energy E where ρ(−∞, E] = l. If we have a large but finite m × m × m crystal with periodic boundary conditions, there are m3l electrons, and in the ground state these fill up the eigenstates of Hm up to an energy Em determined by ρm(−∞, Em] = l. The smallest number E with ρ(−∞, E] = l is called the Fermi energy EF . The set of k ∈ B with En(k) = EF for some n is called the Fermi surface. Observe that this agrees with the definition of F̂λ,R(A, V ) given above. This picture is similar to the elementary discussion of the periodic table based on the hydrogen atom but with the complication of continuum states. We are now in a position to explain why electron conduction is hard in some solids (in- sulators) and easy in others (metals). In the ground state one can use complex conjugation symmetry to prove that there is no net movement of electrons—the expected value of the total momentum is zero. To get flow of electrons one must excite some of the electrons. Usually, a periodic Schrödinger operator has gaps in its spectrum. There is a qualitative difference if EF occurs at the bottom of a gap or not. If EF is at the bottom of a gap, then H has no spectrum in (EF , EF + ε), and there is a discrete amount of energy needed to set up a current. In this case one has an insulator. If EF is not at the bottom of a gap, one 13 has a metal. Of course, if EF is at the bottom of a small gap (ε small), or if EF is not at the bottom of a gap but is fairly close to the bottom of a gap, then one has an intermediate case where the metal/insulator distinction is not sharp (semiconductors). We should also note that in dealing with real solids one must take into account the fact that the solid is not in the ground state but rather in a finite temperature state determined by statistical mechanics. Notice that the gaps in the spectrum are crucial for this theory of insulators versus metals. The Fermi surface plays a fundamental role in this context. We refer the reader to [9] for more details. Typical insulator Typical metal EF Filled band Filled band Lowest energy for conduction Available for conduction Figure 2.3: Energy bands in conductors and insulators. 2.4 Basic properties of Hk(A, V ) Recall that r > d ≥ 2 and let AC and VC be the “complexifications” of A and V, respectively. That is, AC := {(A1, . . . , Ad) | Aj ∈ Lr(Rd/Γ) for 1 ≤ j ≤ d} and VC := Lr/2(Rd/Γ). In this section we prove the following properties of the operator Hk(A, V ). Theorem 2.4.1 (Properties of Hk(A, V )). (a) Let k ∈ Cd and let (A, V ) ∈ AC × VC. Then if λ is not in the spectrum of Hk(A, V ) the resolvent (Hk(A, V )− λ)−1 is a compact operator on L2(Rd/Γ). (b) For k ∈ Rd and (A, V ) ∈ A × V, the operator Hk(A, V ) is self-adjoint in a dense domain DHk(A,V ) ⊂ L2(Rd/Γ). 14 In order to prove this theorem we first introduce some notation and prove a number of propositions. Write H := L2(Rd/Γ) and denote by L (H) the set of all bounded linear operators on H with respect to the operator norm ‖ ·‖ on H. Recall that H is a separable Hilbert space and that the trace class I1 is the set of linear operators A on H such that tr(|A|) < ∞. Here, the operator |A| is the positive square root of A∗A. The trace ideals Ir and its associated norms are defined as Ir := {A ∈ L (H) | tr(|A|r) <∞} and ‖A‖r := (tr(|A|r))1/r for 1 ≤ r <∞, I∞ := {A ∈ L (H) | A is compact} and ‖A‖∞ := ‖A‖ for r =∞. Let µ(A) = {µn(A)}∞n=1 be the singular values of A, that is, the eigenvalues of |A|. One can show that Ir is the set of compact operators on H whose singular values are in lr(N) and ‖A‖r = ‖µ(A)‖lr(N). This is true for any 1 ≤ r ≤ ∞ [15]. The properties of ‖ · ‖r that we need are collected below. Proposition 2.4.2 (Properties of ‖ · ‖r [15]). Let 1 ≤ r < s ≤ ∞ and suppose that A ∈ L (H) and B ∈ Ir. Then: (a) AB ∈ Ir and ‖AB‖r ≤ ‖A‖ ‖B‖r; (b) I1 ⊂ Ir ⊂ Is ⊂ I∞ with ‖ · ‖ ≤ ‖ · ‖s ≤ ‖ · ‖r ≤ ‖ · ‖1; (c) B∗ ∈ Ir and ‖B∗‖r = ‖B‖r. For any ϕ ∈ L2(Rd/Γ) define ϕ̂ : Γ# → C as ϕ̂(b) := (Fϕ)(b) := 1|Γ| ∫ Rd/Γ ϕ(x) e−ib·x dx, where |Γ| := ∫ Rd/Γ dx. Then, ϕ(x) = (F−1ϕ̂)(x) = ∑ b∈Γ# ϕ̂(b) eib·x and ‖ϕ‖L2(Rd/Γ) = |Γ|1/2‖ϕ̂‖l2(Γ#). To simplify the notation we sometimes write Lp and lp in place of Lp(Rd/Γ) and lp(Γ#), respectively. 15 For f ∈ Lr and g ∈ lr define the operators g(−i∇) and f on a suitable domain of L2 as g(−i∇) : ϕ 7→ ∑ b∈Γ# g(b)ϕ̂(b) eib·x and f : ψ 7→ fψ. It turns out that f(x)g(−i∇) and g(−i∇)f(x) are in the trace ideal Ir. More precisely, as in [15, Theorem 4.1], using complex interpolation we prove the following result. Proposition 2.4.3. Let 2 ≤ r ≤ ∞ and suppose that f ∈ Lr(Rd/Γ) and g ∈ lr(Γ#). Then f(x)g(−i∇) ∈ Ir and g(−i∇)f(x) ∈ Ir. Furthermore, ‖f(x)g(−i∇)‖r ≤ |Γ|−1/r ‖f‖Lr‖g‖lr (2.4.1) and ‖g(−i∇)f(x)‖r ≤ |Γ|−1/r ‖f‖Lr‖g‖lr . (2.4.2) Proof. We first prove (2.4.1) and (2.4.2) for r = 2 and r = ∞; then we interpolate using [15, Theorem 2.9]. For r = 2, ‖f(x)g(−i∇)‖22 = tr [(f(x)g(−i∇))∗(f(x)g(−i∇))] = ∑ b∈Γ# ∥∥∥f(x)g(−i∇) |Γ|−1/2 eib·x∥∥∥2 L2 = ∑ b∈Γ# ∥∥∥f(x) |Γ|−1/2 g(b)eib·x∥∥∥2 L2 = |Γ|−1 ∑ b∈Γ# |g(b)|2 ‖f‖2L2 = |Γ|−1 ‖f‖2L2‖g‖2l2 (2.4.3) and ‖g(−i∇)f(x)‖22 = tr [(g(−i∇)f(x))∗(g(−i∇)f(x))] = ∑ b∈Γ# ∥∥∥g(−i∇)f(x) |Γ|−1/2 eib·x∥∥∥2 L2 = ∑ b∈Γ# ∥∥∥∥∥∥g(−i∇) ∑ c∈Γ# f̂(c)|Γ|−1/2 ei(b+c)·x ∥∥∥∥∥∥ 2 L2 = ∑ b∈Γ# ∥∥∥∥∥∥ ∑ c∈Γ# g(b+ c)f̂(c)|Γ|−1/2 ei(b+c)·x ∥∥∥∥∥∥ 2 L2 = ∑ b∈Γ# ∑ c∈Γ# |g(b+ c)|2|f̂(c)|2 = ‖f̂‖2l2‖g‖2l2 = |Γ|−1‖f‖2L2‖g‖2l2 , (2.4.4) while for r =∞, ‖f(x)g(−i∇)ϕ‖2L2 ≤ ‖f‖2L∞‖g(−i∇)ϕ‖2L2 = |Γ| ‖f‖2L∞‖(g(−i∇)ϕ)∧‖2l2 = |Γ| ‖f‖2L∞‖gϕ̂‖2l2 ≤ |Γ| ‖f‖2L∞‖g‖2l∞‖ϕ̂‖2l2 = ‖f‖2L∞‖g‖2l∞‖ϕ‖2L2 16 and ‖g(−i∇)f(x)ϕ‖2L2 = |Γ| ‖(g(−i∇)fϕ)∧‖2l2 = |Γ| ‖g(fϕ)∧‖2l2 ≤ |Γ| ‖g‖2l∞‖(fϕ)∧‖2l2 = ‖g‖2l∞‖fϕ‖2L2 ≤ ‖g‖2l∞‖f‖2L∞‖ϕ‖2L2 . Hence, ‖f(x)g(−i∇)‖ ≤ ‖f‖L∞‖g‖l∞ (2.4.5) and ‖g(−i∇)f(x)‖ ≤ ‖f‖L∞‖g‖l∞ . (2.4.6) The general 2 ≤ r <∞ case now follows by interpolation. Consider the family of norms Φt(A) := (tr(|A|1/t))t for 0 < t ≤ 12 , ‖A‖ for t = 0. One can show that lim t↓0 Φt(A) = Φ0(A) by applying that ‖ · ‖l∞ = lim p→∞ ‖ · ‖lp to the singular values of A. Thus, we have a continuous family of norms. Let z 7→ F1(z) := |Γ|z ei arg f(x) |f |zr(x) ei arg g(−i∇) |g|zr(−i∇) and z 7→ F2(z) := |Γ|z ei arg g(−i∇) |g|zr(−i∇) ei arg f(x) |f |zr(x) be maps from the strip S := {z ∈ C | 0 ≤ Re z ≤ 1/2} to the set of linear operators on L2. Suppose for the moment that f and g are simple functions. We may assume, without loss of generality, that ‖f‖Lr = ‖g‖lr = 1. We shall shortly prove the following property. Proposition 2.4.4. For any ϕ,ψ ∈ L2, and for j ∈ {1, 2}, the function z 7→ 〈ϕ, Fj(z)ψ〉L2 is continuous in S, analytic in its interior, and bounded. Furthermore, in view of (2.4.3) to (2.4.6), for j ∈ {1, 2} we have Φ0(Fj(iy)) ≤ ‖ |Γ|iy ei arg f |f |iyr‖L∞‖ei arg g |g|iyr‖l∞ = 1 17 and Φ1/2 ( Fj ( 1 2 + iy )) ≤ |Γ|− 12 ‖ |Γ| 12 +iy ei arg f |f | r2 +iyr‖L2‖ei arg g |g| r2 +iyr‖l2 = ‖|f | r2 ‖L2‖|g| r 2 ‖l2 = ‖f‖r/2Lr ‖g‖r/2lr = 1. Hence, Fj(iy) ∈ I∞ and Fj ( 1 2 + iy ) ∈ I2 with ‖Fj(iy)‖ ≤ 1 and ∥∥Fj(12 + iy)∥∥2 ≤ 1 for all y ∈ R and j ∈ {1, 2}. By complex interpolation [15, Theorem 2.9], it follows that Fj(z) ∈ I(Re z) with Φ(Re z)(Fj(z)) ≤ 1 for all z ∈ S and j ∈ {1, 2}. In particular, for z = 1/r, |Γ|1/r ‖f(x)g(−i∇)‖r = Φ1/r(F1(1/r)) ≤ 1 and |Γ|1/r ‖g(−i∇)f(x)‖r = Φ1/r(F2(1/r)) ≤ 1. This proves (2.4.1) and (2.4.2) for f and g as above. Since the set of simple functions is dense in Lr(Rd/Γ) and lr(Γ#) for 2 ≤ r ≤ ∞, an approximation argument (using the triangle inequality) completes the proof. We now prove Proposition 2.4.4. Proof of Proposition 2.4.4. Recall that f and g are simple functions and suppose for the moment that ψ is a C∞-function on Rd/Γ with compact support. We first prove that, for all x ∈ Rd/Γ, the function z 7→ (ei arg g(−i∇) |g|zr(−i∇)ψ)(x) is continuous in S, analytic in its interior and bounded. Indeed, let hx,b(z) := ei arg g(b) |g(b)|zrψ̂(b)eib·x. Observe that z 7→ hx,b(z) has the desired properties. Then, since |g(b)|r ≤ ‖g‖rlr ≤ 1 for all b ∈ Γ#, it follows that ∑b∈Γ# |hx,b(z)| ≤∑b∈Γ# |ψ̂(b)| < ∞ (because ψ̂ ∈ l1 since we have assume that ψ is a C∞-function). Thus, the sum (ei arg g(−i∇) |g|zr(−i∇)ψ)(x) = ∑ b∈Γ# hx,b(z) converges uniformly in S (in the uniform norm) by the Weierstrass M-test. Hence it has the same properties as hx,b(z) and the claim follows. 18 Now observe that, for all x ∈ Rd/Γ, the maps z 7→ F1(z)ψ(x) = |Γ|zei arg f(x)|f(x)|zr(ei arg g(−i∇) |g|zr(−i∇)ψ)(x) and z 7→ F2(z)ψ(x) = |Γ|zei arg g(−i∇) |g|zr(−i∇)ei arg f(x)|f(x)|zrψ(x) have the same properties as the above functions. (Note that, in the second map, we can think of ei arg f(x)|f(x)|zrψ(x) as a new ψ̃(x) with the same properties of ψ(x).) Thus, for j ∈ {1, 2}, the function z 7→ 〈ϕ, Fj(z)ψ〉L2 is continuous and bounded in S. This remains true for any ψ ∈ L2 because the set of C∞-functions with compact support is dense in L2. Furthermore, for j ∈ {1, 2} the function z 7→ 〈ϕ, Fj(z)ψ〉L2 is also analytic in the interior of S because, by Cauchy’s Formula and Fubini’s Theorem, 〈ϕ, Fj(z)ψ〉L2 = ∫ Rd/Γ ϕ(x) ( 1 2pii ∫ γ Fj(ξ)ψ(x) ξ − z dξ ) dx = 1 2pii ∫ γ 〈ϕ, Fj(ξ)ψ〉L2 ξ − z dξ, where γ ⊂ S is a closed path around z. This completes the proof of the proposition. The following notation will be used whenever we consider vector-valued quantities. Let X be a Banach space and let A = (A1, . . . , Ad) ∈ X d and B = (B1, . . . , Bd) ∈ X d. Then, ‖A‖X := (‖A1‖2X + · · ·+ ‖Ad‖2X )1/2 and A ·B := A1B1 + · · ·+AdBd. The next three propositions are simple straightforward calculations. Proposition 2.4.5. Let A ∈ (L (H))d and B ∈ I dr . Then A ·B ∈ Ir and B ·A ∈ Ir with ‖A ·B‖r ≤ ‖A‖ ‖B‖r and ‖B ·A‖r ≤ ‖A‖ ‖B‖r. Proof. Using the properties of ‖·‖r (see Proposition 2.4.2) and the Cauchy-Schwarz inequal- ity we have ‖A ·B‖r = ‖A1B1 + · · ·+AdBd‖r ≤ ‖A1B1‖r + · · ·+ ‖AdBd‖r ≤ ‖A1‖ ‖B1‖r + · · ·+ ‖Ad‖ ‖Bd‖r ≤ ‖A‖ ‖B‖r and ‖B ·A‖r = ‖B1A1 + · · ·+BdAd‖r ≤ ‖B1A1‖r + · · ·+ ‖BdAd‖r = ‖A1B1‖r + · · ·+ ‖AdBd‖r ≤ ‖A1‖ ‖B1‖r + · · ·+ ‖Ad‖ ‖Bd‖r ≤ ‖A‖ ‖B‖r, as desired. 19 Proposition 2.4.6. The following inequalities hold:∥∥∥ 1√ I−∆∇ ∥∥∥ ≤ d1/2 and ∥∥∥ 1√ I−∆k ∥∥∥ ≤ |k|. Proof. Let ϕ ∈ L2. Then, for 1 ≤ j ≤ d,∥∥∥ 1√ I−∆ ∂ ∂xj ϕ ∥∥∥2 L2 = |Γ| ∥∥∥ bj√ 1+b2 ϕ̂(b) ∥∥∥2 l2 ≤ |Γ| ‖ϕ̂‖2l2 = ‖ϕ‖2L2 and ∥∥∥ 1√ I−∆kj ϕ ∥∥∥2 L2 = |Γ| ∥∥∥ kj√ 1+b2 ϕ̂(b) ∥∥∥2 l2 ≤ |Γ| |kj |2‖ϕ̂‖2l2 = |kj |2 ‖ϕ‖2L2 . Now observe that, if T is a bounded linear operator and ‖Tϕ‖2L2 ≤ C‖ϕ‖2L2 for all ϕ ∈ L2 then ‖T‖2 ≤ C. Hence, ∥∥∥ 1√ I−∆∇ ∥∥∥ = ( d∑ j=1 ∥∥∥ 1√ I−∆ ∂ ∂xj ϕ ∥∥∥2)1/2 ≤ ( d∑ j=1 1 )1/2 = d1/2 and ∥∥∥ 1√ I−∆k ∥∥∥ = ( d∑ j=1 ∥∥∥ 1√ I−∆kj ϕ ∥∥∥2)1/2 ≤ ( d∑ j=1 |kj |2 )1/2 = |k|, as was to be shown. Proposition 2.4.7. The function g : Γ# → R given by g(b) = (1+b2)−1/2 is in lr for r > d. Proof. There are constants CΓ# and CΓ#,d such that ‖g‖rlr = ∑ b∈Γ# |g(b)|r ≤ CΓ# ∫ Rd (1 + b2)−r/2 db = CΓ# ∫ ∞ 0 (∫ ∂B(0,ρ) dS ) (1 + ρ2)−r/2 dρ = CΓ#,d ∫ ∞ 0 ρd−1(1 + ρ2)−r/2dρ < ∞, since the above integral converges for d− 1− r < −1. Now write Hk(A, V )− λ = I −∆ + u(k, λ) + w(k,A, V ), where ∆ is the Laplace operator in Rd, u(k, λ) := −2ik · ∇+ k2 − λ− I, and w(k,A, V ) := i∇ ·A+ iA · ∇ − 2k ·A+A2 + V. Applying Propositions 2.4.2 to 2.4.7 we prove the following estimates. 20 Lemma 2.4.8. Let r > d. There is a constant C = CΓ,r,d such that∥∥∥ 1√ I−∆w(k,A, V ) 1√ I−∆ ∥∥∥ r ≤ C ((1 + |k|)‖A‖Lr + ‖A‖2Lr + ‖V ‖Lr/2) (a) and ∥∥∥ 1√ I−∆u(k, λ) 1√ I−∆ ∥∥∥ r ≤ C (1 + |k|2 + |λ|). (b) Furthermore, let 0 ≤ ε ≤ r−d2r . There is a constant C = CΓ,r,d,k,λ,A,V such that |〈(u(k, λ) + w(k,A, V ))ϕ,ψ〉L2 | ≤ C(‖(I −∆)(1−ε)/2ϕ‖L2‖(I −∆)1/2ψ‖L2 + ‖(I −∆)1/2ϕ‖L2‖(I −∆)(1−ε)/2ψ‖L2) ≤ 2C ‖(I −∆)1/2ϕ‖L2‖(I −∆)1/2ψ‖L2 (c) for all ϕ,ψ ∈ L2(Rd/Γ). Proof. (a) Write 1√ I −∆ = g(−i∇) with g(b) = 1√ 1 + b2 . Then, using the properties of ‖ · ‖r, in particular that ‖gA‖r = ‖(gA)∗‖r = ‖A∗g∗‖r = ‖A∗g‖r, and Proposition 2.4.6, it follows that ‖gwg‖r = ‖g(i∇ ·A+ iA · ∇ − 2k ·A+A2 + |V |ei arg V )g‖r (by the triangle inequality) ≤ ‖g(∇ ·A)g‖r + 2‖gA · ∇g‖r + 2‖gk ·Ag‖r + ‖gA ·Ag‖r + ‖g|V |1/2ei arg V |V |1/2g‖r (by Proposition 2.4.5) ≤ ‖g∇‖‖Ag‖r + ‖∇g‖ ‖gA‖r + 2‖gk‖ ‖Ag‖r + ‖gA‖ ‖Ag‖r + ‖g|V |1/2‖ ‖|V |1/2g‖r = ‖g∇‖‖Ag‖r + ‖∇g‖ ‖A∗g‖r + 2‖gk‖ ‖Ag‖r + ‖A∗g‖ ‖Ag‖r + ‖|V |1/2g‖ ‖|V |1/2g‖r ≤ (d1/2 + 2|k|)‖Ag‖r + d1/2‖A∗g‖r + ‖A∗g‖r‖Ag‖r + ‖|V |1/2g‖2r . Now, since g ∈ lr for r > d by Proposition 2.4.7, applying Proposition 2.4.3 we obtain ‖gwg‖r ≤ 2(d1/2 + |k|)|Γ|−1/r‖A‖Lr‖g‖lr + |Γ|−2/r‖A‖2Lr‖g‖2lr + |Γ|−2/r‖|V |1/2‖2Lr‖g‖2lr ≤ CΓ,r,d ( (1 + |k|)‖A‖Lr + ‖A‖2Lr + ‖V ‖Lr/2 ) . This proves part (a). 21 (b) The spectrum of g(−i∇)u(k, λ)g(−i∇) is{ 2k · b+ k2 − λ− 1 1 + b2 ∣∣∣∣ b ∈ Γ#} . Hence, ‖gug‖r = ∥∥∥∥2k · b+ k2 − λ− 11 + b2 ∥∥∥∥ lr ≤ ∥∥∥∥ |b|1 + b2 ∥∥∥∥ lr 2|k|+ ∥∥∥∥ 11 + b2 ∥∥∥∥ lr (1 + |k|2 + |λ|) ≤ CΓ,r (1 + |k|2 + |λ|), which proves part (b). (c) Write D = √ I −∆. The condition on ε implies that r(1− ε) ≥ r+d2 > d so that, by Proposition 2.4.7, (1 + b2)−r(1−ε)/2 ∈ l1(Γ#), and hence D−(1−ε) ∈ Ir. Thus, as in part (a), ‖D−1∇ ·AD−(1−ε)‖r ≤ CΓ,r,d ‖A‖Lr , ‖k ·AD−(1−ε)‖r ≤ CΓ,r,d |k| ‖A‖Lr , ‖D−(1−ε)A ·AD−(1−ε)‖r ≤ CΓ,r,d ‖A‖2Lr , ‖D−(1−ε)V D−(1−ε)‖r ≤ CΓ,r,d ‖V ‖Lr/2 . Furthermore, ‖D−1u‖ ≤ sup b∈Γ# ∣∣∣∣2k · b+ k2 − λ− 1√1 + b2 ∣∣∣∣ ≤ 2 (1 + |k|2 + |λ|). Consequently, |〈i∇ ·Aϕ,ψ〉| = |〈D−1i∇ ·AD−(1−ε)D1−εϕ,Dψ〉| ≤ ‖D−1∇ ·AD−(1−ε)D1−εϕ‖L2‖Dψ‖L2 ≤ ‖D−1∇ ·AD−(1−ε)‖ ‖D1−εϕ‖L2‖Dψ‖L2 ≤ ‖D−1∇ ·AD−(1−ε)‖r‖D1−εϕ‖L2‖Dψ‖L2 ≤ CΓ,r,d‖A‖Lr‖D1−εϕ‖L2‖Dψ‖L2 . 22 Similarly, |〈iA · ∇ϕ,ψ〉| ≤ CΓ,r,d‖A‖Lr‖Dϕ‖L2‖D1−εψ‖L2 , |〈2k ·Aϕ,ψ〉| ≤ CΓ,r,d|k| ‖A‖Lr‖D1−εϕ‖L2‖ψ‖L2 , |〈A ·Aϕ,ψ〉| ≤ CΓ,r,d‖A‖2Lr‖D1−εϕ‖L2‖D1−εψ‖L2 , |〈V ϕ, ψ〉| ≤ CΓ,r,d‖V ‖Lr/2‖D1−εϕ‖L2‖D1−εψ‖L2 , |〈uϕ, ψ〉| ≤ CΓ,r,d(1 + |k|2 + |λ|)‖ϕ‖L2‖Dψ‖L2 . Finally, since ‖Dqϕ‖L2 ≤ ‖Dsϕ‖L2 for all 0 ≤ q ≤ s and all ϕ ∈ L2, the above estimates imply inequality (c). The proof of the lemma is complete. Lemma 2.4.8 says that the operators given in (a) and (b) belong to Ir and that the quadratic form in (c) is well-defined on the domain H1(Rd/Γ)×H1(Rd/Γ). Here, the space H1(Rd/Γ) is the usual Sobolev space with norm ‖ · ‖H1(Rd/Γ) = ‖(I −∆)1/2 · ‖L2(Rd/Γ). We now prove the following general property of Sobolev norms. Proposition 2.4.9. Let 0 < r < s < t. Then, given δ > 0 there is a positive constant C = Cδ,r,s,t such that ‖ϕ‖Hs(Rd/Γ) ≤ δ ‖ϕ‖Ht(Rd/Γ) + C ‖ϕ‖Hr(Rd/Γ) for all ϕ ∈ Ht(Rd/Γ). Proof. We first find a constant C > 0 such that (1 + b2)s ≤ δ2(1 + b2)t + C2(1 + b2)r for all b ∈ Γ# or, equivalently, setting y := 1 + b2, such that δ2yt−s + C2 1 ys−r ≥ 1 for all y ≥ 1. Let φ(y) := δ2yt−s + C2 1 ys−r . Then, φ′(y) = 1 ys+1 (δ2(t− s)yt + C2(r − s)yr), so that φ(y) has only one critical point for y > 0, namely, y∗ := [ C2(s− r) δ2(t− s) ] 1 t−r . 23 Since limy→0 φ(y) = limy→∞ φ(y) = ∞, the point y∗ is a point of minimum and, for all y ≥ 1, φ(y) ≥ φ(y∗) = C 2(t−s)t−r [ δ2 [ s− r δ2(t− s) ] t−s t−r + [ s− r δ2(t− s) ] r−s t−r ] = 1 if we choose C = [ δ2 [ s− r δ2(t− s) ] t−s t−r + [ s− r δ2(t− s) ] r−s t−r ] r−t 2(t−s) . Consequently, ‖ϕ‖2Hs = ‖(I −∆)s/2ϕ‖2L2 = |Γ| ‖((I −∆)s/2ϕ)∧‖2l2 = |Γ| ∑ b∈Γ# (1 + b2)s|ϕ̂(b)|2 ≤ δ2|Γ| ∑ b∈Γ# (1 + b2)t|ϕ̂(b)|2 + C2|Γ| ∑ b∈Γ# (1 + b2)r|ϕ̂(b)|2 = δ2‖ϕ‖2Ht + C2‖ϕ‖2Hr ≤ (δ‖ϕ‖Ht + C‖ϕ‖Hr)2 . Taking the square root of both sides of the last expression we obtain the desired inequality. We now show that Hk(A, V ) is a self-adjoint semibounded operator (on a suitable do- main) when k, A and V are real. To prove this property we consider the quadratic form associated to this operator (see [11, §VIII.6]). Proof of Theorem 2.4.1(b). Let D = H1(Rd/Γ) be the domain of √I −∆. By Lemma 2.4.8, the operator (i∇+A− k)2 + V − λ = I −∆ + u(k, λ) + w(k,A, V ) gives a well-defined quadratic form q : D ×D −→ C, (ϕ,ψ) 7−→ 〈(I −∆ + u(k, λ) + w(k,A, V ))ϕ,ψ〉L2 . Furthermore, for ϕ ∈ D, when k, A and V are real, q(ϕ,ϕ) = ‖√I −∆ϕ‖2 + 〈(u(k, λ) + w(k,A, V ))ϕ,ϕ〉L2 ≥ ‖√I −∆ϕ‖2 − C‖(I −∆)(1−ε)/2ϕ‖ ‖√I −∆ϕ‖, where C > 0 is a constant and ‖ · ‖ = ‖ · ‖L2 (to simplify the notation). By Proposition 2.4.9, for any δ > 0 there is a constant Cδ > 0 such that ‖(I −∆)(1−ε)/2 ϕ‖ ≤ δ ‖√I −∆ϕ‖+ Cδ‖ϕ‖ 24 for all ϕ ∈ D. Choosing δ = (2C)−1 we obtain q(ϕ,ϕ) ≥ (1− Cδ)‖√I −∆ϕ‖2 − CCδ‖ϕ‖ ‖ √ I −∆ϕ‖ = 12‖ √ I −∆ϕ‖2 − CCδ‖ϕ‖ ‖ √ I −∆ϕ‖ = 14‖ √ I −∆ϕ‖2 − (CCδ)2‖ϕ‖2 + C‖ϕ‖ ( Cδ √ C‖ϕ‖ − ‖ √ I −∆ϕ‖ 2 √ C‖ϕ‖ )2 ≥ 14‖ √ I −∆ϕ‖2 − (CCδ)2‖ϕ‖2 ≥ −(CCδ)2‖ϕ‖2. Hence, the form q is semibounded and 1 4‖ √ I −∆ϕ‖2 − (CCδ)2‖ϕ‖2 ≤ q(ϕ,ϕ) ≤ C‖ √ I −∆ϕ‖2. (2.4.7) Since D = H1(Rd/Γ) is complete and ‖ϕ‖H1 = ‖ √ I −∆ϕ‖, whenever {ϕn} ⊂ D with ϕn → ϕ in L2 and q(ϕn − ϕm, ϕn − ϕm) → 0 as n,m → ∞, it follows from (2.4.7) that ‖ϕn − ϕm‖H1 → 0 as n,m→∞ and consequently that ϕ ∈ D with q(ϕn − ϕ,ϕn − ϕ)→ 0. This implies that the form q is closed. By [11, Theorem VIII.15], there is a unique self-adjoint semibounded operator Hk(A, V ) densely defined on D ⊂ L2(Rd/Γ) associated to q. Finally, we outline the proof of Theorem 2.4.1(a). Outline of the proof of Theorem 2.4.1(a). To simplify the notation write Hk = Hk(A, V ), and denote by σ(Hk) the spectrum of Hk. If λ 6∈ σ(Hk), the resolvent (Hk − λ)−1 exists and is bounded. This is just the definition of spectrum. We first claim that, if (Hk − λ)−1 is compact for some λ 6∈ σ(Hk), then it is compact for all λ 6∈ σ(Hk). In fact, suppose that λ, λ′ 6∈ σ(Hk). Then, by the resolvent identity we have (Hk − λ′)−1 = (Hk − λ)−1 + (Hk − λ)−1(λ− λ′)(Hk − λ′)−1 = (Hk − λ)−1 [ I + (λ− λ′)(Hk − λ′)−1 ] . The factor [ I + (λ−λ′)(Hk −λ′)−1 ] is a bounded operator. Thus, if (Hk −λ)−1 is compact for some λ /∈ σ(Hk), the above product is compact and consequently (Hk−λ′)−1 is compact for all λ′ /∈ σ(Hk). This shows that is enough to prove part (a) of theorem for a single λ /∈ σ(Hk). This is what we do next. 25 Write Hk = −∆ +Q with Q := i∇ · (A− k) + (A− k) · i∇+ (A− k)2 + V. Then, if λ /∈ σ(Hk), and if |λ| is sufficiently large, the operator( I + 1√−∆− λ Q 1√−∆− λ )−1 exists and is bounded. Furthermore, we can choose the λ so that 1√−∆−λ exists, and similarly as in the proof of Lemma 2.4.8(c), we can prove that this operator is compact. Thus we can write (Hk − λ)−1 = 1√−∆− λ 1 I + 1√−∆−λ Q 1√−∆−λ 1√−∆− λ, and similarly as above conclude that (Hk − λ)−1 is compact for a suitable λ /∈ σ(Hk). In view of the above remark, this implies the statement for all λ /∈ σ(Hk) and completes the outline of the proof. 2.5 The complex analytic structure of the spectrum For complex-valued potentials (A, V ) ∈ AC × VC and for k ∈ C2 the problem Hϕn,k = En(k)ϕn,k, ϕn,k(x+ γ) = eik·γϕn,k(x) for all x ∈ Rd and all γ ∈ Γ is no longer self-adjoint. Its spectrum, however, remains discrete. It is a sequence of eigenvalues in the complex plane. From the above boundary condition it is easy to see that for each n ≥ 1 the function k 7→ En(k) remains periodic with respect to Γ#. Furthermore, the transformation Uk is no longer unitary but it is still bounded and invertible and it still preserves the spectrum. That is, we can still rewrite this problem in the form Hkψn,k = En(k)ψn,k for ψn,k ∈ L2(Rd/Γ) without modifying the eigenvalues. Thus, as above we define the (complex) “lifted” Fermi surface of (A, V ) with energy λ ∈ C to be F̂λ(A, V ) := {k ∈ Cd | (Hk(A, V )− λ)ψ = 0 for some ψ ∈ DHk(A,V ) \ {0}}. 26 We shall prove below that this surface has the following (well-known) property. Theorem 2.5.1 ([3]). There exists an analytic function F on Cd ×C×AC ×VC such that F̂λ(A, V ) = {k ∈ Cd | F (k, λ,A, V ) = 0}. In particular, for k, A and V real, λ ∈ Spec (Hk(A, V )) if and only if F (k, λ,A, V ) = 0. Furthermore, such an analytic function F is given by (2.5.1). To prove this theorem we follow [3]. Another proof of a similar statement can be found in [8, Theorem 4.4.2]. We shall use the definition and some properties of the regularized determinant on Ir (see [15, Chapter 9]). Proof. Since Lp(Rd/Γ) ⊃ Lq(Rd/Γ) for all 1 ≤ p < q, we may assume, without loss of generality, that r ≤ d+ 1. Then, since ‖ · ‖d+1 ≤ ‖ · ‖r, Lemma 2.4.8 implies that F (k, λ,A, V ) := detd+1 ( I + 1√ I−∆u(k, λ) 1√ I−∆ + 1√ I−∆w(k,A, V ) 1√ I−∆ ) (2.5.1) is a well-defined function on Cd × C × AC × VC. Here detd+1(I + B) is the regularized determinant for B ∈ Id+1. This function is Fréchet differentiable [15] and hence analytic on its domain [1, Theorem 14.13]. Analyticity may also be proved as in [6] by approximating B by a sequence of finite rank operators. Now, let µ be a complex number that is not in the spectrum of Hk(A, V ). Then, by Theorem 2.4.1(a), the resolvent (Hk(A, V )−µ)−1 is a compact operator on L2(Rd/Γ). Thus, the spectrum of (Hk(A, V )−µ)−1 is discrete, and so is the spectrum of Hk(A, V ). Therefore, λ ∈ Spec(Hk(A, V )) if and only if there exists ψ ∈ DHk(A,V ) ⊂ D such that (Hk(A, V )− λ) 1√ I −∆ √ I −∆ψ = 0. This is the case if and only if 1√ I−∆(Hk(A, V )−λ) 1√I−∆ is not invertible. Since this operator is the sum of the identity and a compact operator, it fails to be invertible if and only if it has a nontrivial kernel. By [15, Theorem 9.2], this is the case if and only if F (k, λ,A, V ) = 0. 27 2.6 Gauge invariance In this section we outline the proof of the following (well-known) properties. Theorem 2.6.1 (Gauge invariance). For 1 ≤ j ≤ d, let Aj ∈ C1(Rd/Γ), V ∈ C0(Rd/Γ), and Ψ ∈ C2(Rd/Γ). Then DHk(A,V ) = D−∆+I and eiΨ : DHk(A,V ) → DHk(A,V ). Furthermore: (a) Ker(Hk(A, V )− λ) 6= {0} if and only if Ker(e−iΨHk(A, V )eiΨ − λ) 6= {0}; (b) e−iΨHk(A, V )eiΨ = Hk(A−∇Ψ, V ); (c) F̂λ(A, V ) = F̂λ(A−∇Ψ, V ). Outline of the Proof. We first prove parts (a) to (c). (a) Consider the linear transformation eiΨ : ϕ(x) 7−→ eiΨ(x)ϕ(x) acting on DHk(A,V ). Since the function Ψ is bounded, it is clear that eiΨ is a bounded operator with bounded inverse on L2(Rd/Γ) given by e−iΨ. Define HΨk (A, V ) := e −iΨHk(A, V )eiΨ and DHΨk (A,V ) := e −iΨDHk(A,V ). Observe that, by hypothesis, DHΨk (A,V ) = DHk(A,V ). Furthermore, we claim that Hk(A, V )ψ = λψ for ψ ∈ DHk(A,V ) with ψ 6= 0, if and only if HΨk (A, V )ϕ = λϕ, where ϕ = e−iΨψ with ϕ ∈ DHk(A,V ) and ϕ 6= 0. Indeed, observe that the former equation implies the last one, 0 = e−iΨ0 = e−iΨ(Hk(A, V )− λ)ψ = e−iΨ(Hk(A, V )− λ)eiΨe−iΨψ = (e−iΨHk(A, V )eiΨ − λ)e−iΨψ = (HΨk (A, V )− λ)ϕ, where ϕ 6= 0 since ψ 6= 0, and that a similar calculation (in the reverse order) implies the converse statement. This proves part (a). 28 (b) To simplify the notation write A′ = A−∇Ψ. Observe that, formally (without worry about domains and weak derivatives), e−iΨHk(A, V )eiΨϕ = e−iΨ((i∇+A− k)2 + V )eiΨϕ = e−iΨ(i∇+A− k) · eiΨ(−(∇Ψ) + i∇+A− k))ϕ+ V ϕ = −(∇Ψ) +A− k) · (−(∇Ψ) + i∇+A− k)ϕ+ i∇ · (−(∇Ψ) + i∇+A− k)ϕ+ V ϕ = (k2 − 2ik · ∇ −∆ +A′ · (i∇− k) + (i∇− k) ·A′ + (A′)2 + V )ϕ = ((i∇− k)2 + (i∇− k) ·A′ +A′ · (i∇− k) + (A′)2 + V )ϕ = ((i∇+A′ − k)2 + V )ϕ. However, since e−iΨDHk(A,V ) = DHk(A,V ), we claim that this calculation can be rigorously justified indeed. Assuming this we have e−iΨHk(A, V )eiΨ = Hk(A−∇Ψ, V ), which proves part (b). (c) As a consequence of parts (a) and (b) we obtain F̂λ(A, V ) = {k ∈ Cd | (Hk(A, V )− λ)ψ = 0 for some ψ ∈ DHk(A,V ) \ {0}} = {k ∈ Cd | (e−iΨHk(A, V )eiΨ − λ)ϕ = 0 for some ϕ ∈ DHk(A,V ) \ {0}} = {k ∈ Cd | (Hk(A−∇Ψ, V )− λ)ϕ = 0 for some ϕ ∈ DHk(A−∇Ψ,V ) \ {0}} = F̂λ(A−∇Ψ, V ), as desired. We now outline the proof of the first part of the theorem. First observe that, since D−∆+I = H2(Rd/Γ), and H2(Rd/Γ) is dense in L2(Rd/Γ), the set D−∆+I is dense in DHk(A,V ). Furthermore, it is easy to verify that D−∆+I ⊂ DHk(A,V ). Thus, if we can prove that Hk(A, V ) with domain D−∆+I is closed, then it follows that DHk(A,V ) = D−∆+I . That is, we can choose D−∆+I as a domain for Hk(A, V ). Furthermore, since Ψ ∈ C2(Rd/Γ), it is clear then that eiΨ maps DHk(A,V ) to DHk(A,V ). Therefore, to conclude the proof of the theorem it suffices to show that Hk(A, V ) is closed in D−∆+I . 29 The operator Hk(A, V ) is closed in D−∆+I if its graph is closed in D−∆+I × D−∆+I . Equivalently, its graph is closed if for any {ϕn} ⊂ D−∆+I such that limn→∞ ϕn =: ϕ exists and limn→∞Hk(A, V )ϕn =: ψ exists, then ϕ ∈ D−∆+I andHk(A, V )ϕ = ψ with ψ ∈ D−∆+I . To prove that Hk(A, V ) is closed write Hk(A, V ) = −∆ +Q with Q := i∇ · (A− k) + (A− k) · i∇+ (A− k)2 + V, and assume that limn→∞ ϕn =: ϕ and limn→∞Hk(A, V )ϕn =: ψ exist. Then, for a suitable constant λ, (Hk(A, V ) + λ)ϕn = (−∆ + λ+Q)ϕn = [ I +Q(−∆ + λ)−1](−∆ + λ)ϕn, where the operator Q(−∆ + λ)−1 is bounded. Furthermore, we can choose λ sufficiently large so that the operator norm of Q(−∆ + λ)−1 is strictly less than 1. Consequently, the operator [ I + Q(−∆ + λ)−1] is bounded and has a bounded inverse. Since by hypothesis {(Hk(A, V ) +λ)ϕn} converges, we conclude that limn→∞(−∆ +λ)ϕn exists. It follows then that ϕ ∈ D−∆+I and (−∆+λ)ϕ ∈ D−∆+I because −∆+λ is closed in D−∆+I (we are using this fact without proof). Finally, since [ I +Q(−∆ + λ)−1] is bounded and has a bounded inverse, it follows that limn→∞(Hk(A, V ) + λ)ϕn is in D−∆+I . All this together shows that Hk(A, V ) is closed in D−∆+I and completes the proof of the theorem. 30 Chapter 3 Asymptotics for Fermi curves 3.1 Fermi curves Below we define the Fermi curves and briefly describe some of its properties. Let Γ be a lattice in R2 and let A1, A2 and V be real-valued functions in L2(R2/Γ). Set A := (A1, A2) and define the operator H(A, V ) := (i∇+A)2 + V acting on L2(R2), where ∇ is the gradient operator in R2. For k ∈ R2 consider the following self-adjoint eigenvalue-eigenvector problem with boundary conditions, H(A, V )ϕ = λϕ, ϕ(x+ γ) = eik·γϕ(x) for all x ∈ R2 and all γ ∈ Γ. The spectrum of this problem is discrete. It consists of a sequence of real eigenvalues E1(k,A, V ) ≤ E2(k,A, V ) ≤ · · · ≤ En(k,A, V ) ≤ · · · For each integer n ≥ 1 the eigenvalue En(k,A, V ) defines a continuous function of k. From the above boundary condition it is easy to see that this function is periodic with respect to the dual lattice Γ# := {b ∈ R2 | b · γ ∈ 2piZ for all γ ∈ Γ}, 31 where b · γ is the usual scalar product on R2. It is customary to refer to k as the crystal momentum and to En(k,A, V ) as the n-th band function. The corresponding normalized eigenfunctions ϕn,k are called Bloch eigenfunctions. Let Uk be the unitary transformation on L2(R2) that acts as Uk : ϕ(x) 7−→ eik·xϕ(x). By applying this transformation we can rewrite the above problem and put the boundary conditions into the operator. Indeed, if we define Hk(A, V ) := U−1k H(A, V )Uk and ψ := U −1 k ϕ, then the above problem is unitary equivalent to Hk(A, V )ψ = λψ, ψ(x+ γ) = ψ(x) for all x ∈ R2 and all γ ∈ Γ, or, using a more compact notation, Hk(A, V )ψ = λψ for ψ ∈ L2(R2/Γ). To see that these problems are equivalent we proceed (formally) as follows. On the one hand, from the original problem and using the above transformation we find that 0 = U−1k 0 = U −1 k (H(A, V )− λ)ϕ = U−1k (H(A, V )− λ)UkU−1k ϕ = (U−1k H(A, V )Uk − λ)U−1k ϕ = (Hk(A, V )− λ)ψ and ψ(x+ γ) = (U−1k ϕ)(x+ γ) = e −ik·(x+γ)ϕ(x+ γ) = e−ik·(x+γ)eik·γϕ(x) = e−ik·xϕ(x) = (U−1k ϕ)(x) = ψ(x). On the other hand, by a similar computation (in the reverse order), using the last two equalities and the above transformation we derive the original problem. This (formally) implies unitary equivalence. Furthermore, a simple (formal) calculation shows that Hk(A, V ) = (i∇+A− k)2 + V. 32 In fact, Hk(A, V )ψ = U−1k H(A, V )Ukψ = e−ik·x [ (i∇+A)2 + V ]eik·xψ = e−ik·x [ (i∇+A) · eik·x(−kψ + i∇ψ +Aψ)]+ V ψ = (−k +A) · (−k + i∇+A)ψ + i∇ · (−k + i∇+A)ψ + V ψ = [ (i∇+A− k)2 + V ]ψ. Of course, the unitary transformation Uk preserves self-adjointness and does not change the spectrum {En(k,A, V )}∞n=1. The real “lifted” Fermi curve of (A, V ) with energy λ ∈ R is defined as F̂λ,R(A, V ) := {k ∈ R2 | En(k,A, V ) = λ for some n ≥ 1}. Equivalently, F̂λ,R(A, V ) = {k ∈ R2 | (Hk(A, V )− λ)ϕ = 0 for some ϕ ∈ DHk(A,V ) \ {0}}, where DHk(A,V ) ⊂ L2(R2/Γ) denotes the (dense) domain of Hk(A, V ). The adjective “lifted” indicates that F̂λ,R(A, V ) is a subset of R2 rather than R2/Γ#. As we may replace V by V −λ, we only discuss the case λ = 0 and write F̂R(A, V ) in place of F̂0,R(A, V ) to simplify the notation. Furthermore, since Hk(A, V ) = Hk−Â(0)(A − Â(0), V ), if we perform the change of coordinates k → k + Â(0) and redefine A − Â(0) → A we may assume, without loss of generality, that Â(0) = 1 |Γ| ∫ R2/Γ A(x) dx = 0. The dual lattice Γ# acts on R2 by translating k 7→ k + b for b ∈ Γ#. This action maps F̂R(A, V ) to itself because for each n ≥ 1 the function k 7→ En(k,A, V ) is periodic with respect to Γ#. In other words, the real lifted Fermi curve “is periodic” with respect to Γ#. Define FR(A, V ) := F̂R(A, V )/Γ#. We call FR(A, V ) the real Fermi curve of (A, V ). It is a curve in the torus R2/Γ#. The above definitions and the real Fermi curve have physical meaning. It is useful and interesting, however, to study the “complexification” of these curves. Knowledge about the complexified curves may provide information about the real counterparts. 33 For complex-valued functions A1, A2 and V in L2(R2/Γ) and for k ∈ C2 the above problem is no longer self-adjoint. Its spectrum, however, remains discrete. It is a sequence of eigenvalues in the complex plane. From the boundary condition in the original problem it is easy to see that the family of functions k 7→ En(k,A, V ) remains periodic with respect to Γ#. Furthermore, the transformation Uk is no longer unitary but it is still bounded and invertible and it still preserves the spectrum, that is, we can still rewrite the original problem in the form Hk(A, V )ψ = λψ for ψ ∈ L2(R2/Γ) without modifying the eigenvalues. Thus, it makes sense to define F̂(A, V ) := {k ∈ C2 | Hk(A, V )ϕ = 0 for some ϕ ∈ DHk(A,V ) \ {0}} and F(A, V ) := F̂(A, V )/Γ#. We call F̂(A, V ) and F(A, V ) the (complex) “lifted” Fermi curve and the (complex) Fermi curve, respectively. When there is no risk of confusion we shall refer to either simply as Fermi curve. Let {γ1, γ2} be a basis of Γ and set C∗ := C \ {0}. Define the exponential map E as E : C2 −→ C∗ × C∗, k 7−→ (eik·γ1 , eik·γ2). This map is holomorphic and the pair (C2, E) is a covering space of C∗×C∗. In fact, every point of C∗ × C∗ has an open neighbourhood W ⊂ C∗ × C∗ such that the inverse image of W under E is a disjoint union of open sets Uj ⊂ C2, with the map E sending each Uj homeomorphically onto W . If we recall that F̂(A, V ) is invariant under the action of Γ# and observe that b · γ1 ∈ 2piZ and b · γ2 ∈ 2piZ for all b ∈ Γ#, it is not difficult to see that E (F̂(A, V )) ∼= F(A, V ). That is, up to the isomorphism J : E (F̂(A, V )) −→ F(A, V ), (eik·γ1 , eik·γ2) 7−→ [k], 34 where [k] denotes a point (or equivalence class) in R2/Γ#, the curve F(A, V ) is the image of F̂(A, V ) under the exponential map. We sometimes assume that the isomorphism J is understood and simply write E (F̂(A, V )) = F(A, V ). Alternatively, we could have used this expression to define F(A, V ) (and avoid talking about the isomorphism J). 3.2 The free Fermi curve When the potentials A and V are zero the curve F̂(A, V ) can be found explicitly. In this section we collect some properties of this curve. For ν ∈ {1, 2} and b ∈ Γ# set Nb,ν(k) := (k1 + b1) + i(−1)ν(k2 + b2), Nν(b) := {k ∈ C2 | Nb,ν(k) = 0}, Nb(k) := Nb,1(k)Nb,2(k), Nb := N1(b) ∪N2(b), θν(b) := 12((−1)νb2 + ib1). Observe that Nν(b) is a line in C2. The free lifted Fermi curve is an union of these lines. Here is the precise statement. Theorem 3.2.1 (The free Fermi curve). The curve F̂(0, 0) is the locally finite union ⋃ b∈Γ# ν∈{1,2} Nν(b). In particular, the curve F(0, 0) is a complex analytic curve in C2/Γ# and F(0, 0) ∼= E(N0). Before we prove this theorem we shall prove some simple properties of the lines Nν(b). Proposition 3.2.2 (Properties of Nν(b)). Let ν ∈ {1, 2} and let b, c, d ∈ Γ#. Then: (a) Nν(b) ∩Nν(c) = ∅ if b 6= c ; (b) dist(Nν(b),Nν(c)) = 1√2 |b− c|; 35 (c) N1(b) ∩N2(c) = {(iθ1(c) + iθ2(b), θ1(c)− θ2(b))}; (d) the map k 7→ k + d maps Nν(b) to Nν(b− d); (e) the map k 7→ k + d maps N1(b) ∩N2(c) to N1(b− d) ∩N2(c− d). Proof. (a) By contradiction. Suppose that Nν(b) ∩ Nν(c) is not empty. Then there is at least one k ∈ C2 such that k1 + b1 + i(−1)ν(k2 + b2) = 0, k1 + c1 + i(−1)ν(k2 + c2) = 0. This is true if and only if b1 − c1 + i(−1)ν(b2 − c2) = 0. But this is impossible because b 6= c and b1, b2, c1 and c2 are real numbers. Thus, the intersection Nν(b) ∩Nν(c) is empty. This proves part (a). (b) Let k ∈ Nν(b) and k′ ∈ Nν(c). Then, k1 + b1 + i(−1)ν(k2 + b2) = 0, k′1 + c1 + i(−1)ν(k′2 + c2) = 0. Write d := c− b. Hence, using the above equations we obtain dist(Nν(b),Nν(c)) = inf{|k − k′| | k ∈ Nν(b) and k′ ∈ Nν(c)} = inf{|(k1 − k′1, k2 − k′2)| | k ∈ Nν(b) and k′ ∈ Nν(c)} = inf{|(c1 − b1 + i(−1)ν(c2 − b2 + k′2 − k2), k2 − k′2)| | k2, k′2 ∈ C} = inf{|(d1 + i(−1)ν(d2 + z),−z)| | z ∈ C} = inf{(d2 − 2 Re(i(−1)νd1z̄ − d2z̄) + 2|z|2)1/2 | z ∈ C} = inf{(d2 + 2(d2x− (−1)νd1y) + 2(x2 + y2))1/2 | x, y ∈ R} = inf {( 1 2d 2 + ( 1√ 2 (d2,−(−1)νd1) + √ 2 (x, y) )2)1/2 ∣∣ (x, y) ∈ R2} = 1√ 2 |d| = 1√ 2 |b− c|, as claimed. (c) Let k ∈ N1(b) ∩N2(c). Then, k1 + b1 − i(k2 + b2) = 0, k1 + c1 + i(k2 + c2) = 0, 36 which hold if and only if k1 = 12 (−b1 − c1 + i(b2 − c2)) = iθ1(c) + iθ2(b), k2 = 12 (−b2 − c2 + i(c1 − b1)) = θ1(c)− θ2(b). This proves part (c). (d) Observe that Nν(b) + d = {k + d ∈ C2 | k1 + b1 + i(−1)ν(k2 + b2) = 0} = {k′ ∈ C2 | k′1 + b1 − d1 + i(−1)ν(k′2 + b2 − d2) = 0} = {k′ ∈ C2 | Nb−d,ν(k′) = 0} = Nν(b− d). This shows that the map k 7→ k + d maps Nν(b) to Nν(b− d), as desired. (e) Similarly as in part (d), the statement of part (e) follows from the equality N1(b) ∩N2(c) + d = {( 1 2 (−b1 − c1 + i(b2 − c2)) + d1, 12 (−b2 − c2 + i(c1 − b1)) + d2 )} = {( 1 2 (−(b1 − d1)− (c1 − d1) + i((b2 − d2)− (c2 − d2))) , 1 2 (−(b2 − d2)− (c2 − d2) + i((c1 − d1)− (b1 − d1))) )} = N1(b− d) ∩N2(c− d). The proof of the proposition is complete. We now prove Theorem 3.2.1. Proof of Theorem 3.2.1. For all k ∈ C2 the functions {eib·x | b ∈ Γ#} form a complete set of eigenfunctions for Hk(0, 0) in L2(R2/Γ) satisfying Hk(0, 0)eib·x = (i∇− k)2eib·x = (b+ k)2eib·x = Nb(k)eib·x. Hence, F̂(0, 0) = {k ∈ C2 | Nb(k) = 0 for some b ∈ Γ#} = ⋃ b∈Γ# Nb = ⋃ b∈Γ# ν∈{1,2} Nν(b). This is the desired expression for F̂(0, 0). We next prove that this union is locally finite. First observe that, since the lattice Γ# is discrete, there is a constant C > 0 such that |b − c| ≥ C for all distinct elements b, c ∈ Γ#. Thus, by Proposition 3.2.2(b), the distance 37 between any two distinct lines Nν(b) and Nν(c) is bounded below by dist(Nν(b),Nν(c)) = 1√ 2 |b− c| ≥ C√ 2 > 0. Now, let U be an open bounded subset of C2. We claim that only a finite number of lines Nν(b) can intersect U . Indeed, suppose this is not the case. Then for at least one ν ∈ {1, 2} there is an infinite number of lines Nν(b) crossing U . In particular, there is at least one point of each of these lines inside U . By the above inequality, all these points are apart from each other by at least C/ √ 2. Since we have an infinite number of such points inside U this implies that U is unbounded. But this is a contradiction. Therefore, only a finite number of lines Nν(b) can intersect U . Consequently, given U there exits a finite set B ⊂ Γ# such that F̂(0, 0) ∩ U = U ∩ ⋃ b∈Γ# Nb = ⋃ b∈Γ# Nb ∩ U = ⋃ b∈B Nb ∩ U = {k ∈ U | Nb(k) = 0 for b ∈ B}. That is, the union F̂(0, 0) is locally finite. Furthermore, the curve F̂(0, 0) is locally the zero set of a finite number of polynomials Nb(k) and hence it is a complex analytic curve in C2. Clearly the same conclusion holds for F(0, 0) in C2/Γ#. Indeed, consider F(0, 0) ∩W for some open bounded subset W in C2/Γ#. Then if we embed this set in F̂(0, 0) ⊂ C2 “around some b ∈ Γ# \ {0}”, we obtain a finite number of defining equations for F(0, 0) ∩W . This can be properly done by exploiting the covering property F(0, 0) ∼= E(N0). To prove this relation we proceed as follows. Let N0 − b be the translation of N0 by −b ∈ Γ#. Then, by Proposition 3.2.2(d) we have Nb = N0 − b for all b ∈ Γ#. Hence, F(0, 0) ∼= E(F̂(0, 0)) = E( ⋃ b∈Γ# Nb ) = ⋃ b∈Γ# E(Nb) = ⋃ b∈Γ# E(N0 − d) = E(N0) = E(N1(0) ∪N2(0)) = E(N1(0)) ∪ E(N2(0)) = E({(ik2, k2) | k2 ∈ C}) ∪ E({(−ik2, k2) | k2 ∈ C}) = { (eik2(i,1)·γ1 , eik2(i,1)·γ2) ∣∣ k2 ∈ C} ∪ {(eik2(−i,1)·γ1 , eik2(−i,1)·γ2) ∣∣ k2 ∈ C}. In particular this shows that F(0, 0) ∼= E(N0) and completes the proof of the theorem. Let us briefly describe what the free Fermi curve looks like. In the Figure 3.1 there is a sketch of the set of (k1, k2) ∈ F̂(0, 0) for which both ik1 and k2 are real, for the case where 38 the lattice Γ# has points over the coordinate axes, that is, it has points of the form (b1, 0) and (0, b2). Observe that, in particular, Proposition 3.2.2 yields N1(0) ∩N2(b) = {(iθ1(b), θ1(b))}, N1(−b) ∩N2(0) = {(iθ2(−b), θ2(b))}, the map k 7→ k + b maps N1(0) ∩N2(b) to N1(−b) ∩N2(0). Recall that points in F̂(0, 0) that differ by elements of Γ# correspond to the same point in F(0, 0). Thus, in the sketch on the left, we should identify the lines k2 = −b2/2 and k2 = b2/2 for all b ∈ Γ# with b2 6= 0, to get a pair of helices climbing up the outside of a cylinder, as illustrated by the figure on the right. The helices intersect each other twice on each cycle of the cylinder—once on the front half of the cylinder and once on the back half. Hence, viewed as a “manifold” (with singularities), the pair of helices are just two copies of R with points that corresponds to intersections identified. We can use k2 as a coordinate in each copy of R and then the pairs of identified points are k2 = b2/2 and k2 = −b2/2 for all b ∈ Γ# with b2 6= 0 (see Figure 3.2). N2(0) N2(b) N1(−b) N1(0) N1(b)N2(−b) k2 ik1 k2 ik1 Figure 3.1: Sketch of F̂(0, 0) and F(0, 0) when both ik1 and k2 are real. So far we have only considered k2 real. The full F̂(0, 0) is just two copies of C with k2 as a coordinate in each copy, provided we identify the points θ1(b) = 12(−b2 + ib1) (in the first copy) and θ2(b) = 12(b2 + ib1) (in the second copy) for all b ∈ Γ# with b2 6= 0 (see Figure 3.3). 39 RR Figure 3.2: The pair of helices as a “manifold”. Figure 3.3: The full F̂(0, 0): two copies of C with θ1(b) and θ2(b) identified for b2 6= 0. To conclude this section we give a (global) defining equation for F̂(0, 0). For b ∈ Γ# set Rb(k) := exp [ −2k · b+ k 2 − 1 1 + b2 + 1 2 [ 2k · b+ k2 − 1 1 + b2 ]2] . We have the following proposition. Proposition 3.2.3 (Global defining equation for F̂(0, 0)). The following equality holds: F̂(0, 0) = { k ∈ C2 ∣∣∣∣∣ ∏ b∈Γ# (k + b)2 1 + b2 Rb(k) = 0 } , where the infinite product inside brackets converges to an entire function on C2. Before we prove this proposition we remark that the above product is analogous to the canonical product associated to a sequence of complex numbers {zn}. Such product defines an entire function on C which has a zero at each point zn (see [13, p 302]). 40 Proof. We give only a sketch of the proof. By Theorem 2.5.1 we have F̂(0, 0) = {k ∈ C2 | F (k, 0, 0, 0) = 0} with F (k, 0, 0, 0) = det3 ( I + 1√ I−∆u(k, 0) 1√ I−∆ ) . Write T := 1√ I−∆u(k, 0) 1√ I−∆ = 1√ I−∆(−2ik · ∇+ k 2 − I) 1√ I−∆ and set R3(T ) := (I + T ) exp 2∑ j=1 (−1)j j T j − I. The eigenvalues of R3(T ), which we denote by {λb(R3(T ))}b∈Γ# , are given by λb(R3(T )) = (k + b)2 1 + b2 Rb(k)− 1. Hence (see [15, Chapter 9] for details), F (k, 0, 0, 0) = det3(I + T ) = det(I +R3(T )) = ∏ b∈Γ# (1 + λb(R3(T ))) = ∏ b∈Γ# (k + b)2 1 + b2 Rb(k). Furthermore, according to Theorem 2.5.1 the function F (k, 0, 0, 0) is analytic on C2. This proves the proposition. 3.3 The ε-tubes about the free Fermi curve We now introduce real and imaginary coordinates in C2 and define ε-tubes about the free Fermi curve. We derive some properties of the ε-tubes as well. For k ∈ C2 write k1 = u1 + iv1 and k2 = u2 + iv2, where u1, u2, v1 and v2 are real numbers. Then, Nb,ν(k) = (k1 + b1) + i(−1)ν(k2 + b2) = i(v1 + (−1)ν(u2 + b2))− (−1)ν(v2 − (−1)ν(u1 + b1)), 41 so that |Nb,ν(k)| = |v + (−1)ν(u+ b)⊥|, where (y1, y2)⊥ := (y2,−y1). Observe that (y⊥)⊥ = −y and |y⊥| = |y|. Furthermore, for any real number λ we have (λy)⊥ = λy⊥. Since Nb(k) = Nb,1(k)Nb,2(k), it follows that Nb(k) = 0 if and only if v − (u+ b)⊥ = 0 or v + (u+ b)⊥ = 0. Let 2Λ be the length of the shortest nonzero “vector” in Γ#. Then there is at most one b ∈ Γ# with |v + (u + b)⊥| < Λ and at most one b ∈ Γ# with |v − (u + b)⊥| < Λ. Indeed, suppose there is another b′ 6= b such that |v + (u+ b′)⊥| < Λ or |v − (u+ b′)⊥| < Λ. Then, |b− b′| = |(b− b′)⊥| = |v ± (u+ b)⊥ − (v ± (u+ b′)⊥)| < Λ + Λ = 2Λ, which contradicts the definition of Λ. Thus, there is no such b′ (see Figure 3.4). Λ v⊥ − u v⊥ + u Figure 3.4: On the left: definition of Λ. On the right: possible configuration. Let ε be a constant satisfying 0 < ε < Λ 6 . For ν ∈ {1, 2} and b ∈ Γ# define the ε-tube about Nν(b) as Tν(b) := {k ∈ C2 | |Nb,ν(k)| = |v + (−1)ν(u+ b)⊥| < ε}, 42 and the ε-tube about Nb = N1(b) ∪N2(b) as Tb := T1(b) ∪ T2(b). Since (v + (u + b)⊥) + (v − (u + b)⊥) = 2v, at least one of the factors |v + (u + b)⊥| or |v − (u + b)⊥| in |Nb(k)| must always be greater or equal to |v|. If k 6∈ Tb both factors are also greater or equal to ε. If k ∈ Tb one factor is bounded by ε and the other must lie within ε of |2v|. Thus, k 6∈ Tb =⇒ |Nb(k)| ≥ ε|v|, (3.3.1) k ∈ Tb =⇒ |Nb(k)| ≤ ε(2|v|+ ε). (3.3.2) The pairwise intersection T b ∩ T b′ is compact whenever b 6= b′. Here T b denotes the closure of Tb. Indeed, to prove this first observe that, the intersection T ν(b) ∩ T ν(b′) is empty if b 6= b′ because, if it were not, we would have |v + (−1)ν(u+ b)⊥ − v − (−1)ν(u+ b′)⊥| ≤ 2ε < Λ 3 , which contradicts |v + (−1)ν(u+ b)⊥ − v − (−1)ν(u+ b′)⊥| = |b− b′| ≥ 2Λ, which is certainly true according to the definition of Λ. Furthermore, if k ∈ T 1(b) ∩ T 2(b′) then |u+ 12(b+ b′)| = 12 |v − (u+ b)⊥ − v − (u+ b′)⊥| ≤ ε2 + ε2 = ε, |v − 12(b− b′)⊥| = 12 |v − (u+ b)⊥ + v + (u+ b′)⊥| ≤ ε2 + ε2 = ε, which defines a compact set. Thus, the intersection T b ∩ T b′ = (T 1(b) ∩ T 2(b′)) ∪ (T 2(b) ∩ T 1(b′)) (3.3.3) is compact. Finally, it is easy to see that T b ∩ T b′ ∩ T b′′ is empty for all distinct elements b, b′, b′′ ∈ Γ#. In fact, in view of (3.3.3), T b ∩ T b′ ∩ T b′′ = [ T 1(b) ∩ T 2(b′) ∩ T 1(b′′) ] ∪ [T 1(b) ∩ T 2(b′) ∩ T 2(b′′)] ∪ [T 2(b) ∩ T 1(b′) ∩ T1(b′′)] ∪ [T 2(b) ∩ T 1(b′) ∩ T2(b′′)] = ∅ ∪∅ ∪∅ ∪∅ = ∅. 43 ik1 k2 T2(0) T1(0) T2(−b) T1(b) T2(b) T1(−b) Figure 3.5: The ε-tubes about the free “lifted” Fermi curve. If a point k belongs to the free Fermi curve the function Nb(k) vanishes for some b ∈ Γ#. To conclude this section we give a lower bound for this function when (b, k) is away from the zero set. Proposition 3.3.1 (Lower bound for |Nb(k)|). (a) If |b+ u+ v⊥| ≥ Λ and |b+ u− v⊥| ≥ Λ, then |Nb(k)| ≥ Λ2 (|v|+ |u+ b|). (b) If |v| > 2Λ and k ∈ T0, then |Nb(k)| ≥ Λ2 (|v| + |u + b|) for all b 6= 0 but at most one b 6= 0. This exceptional b̃ obeys |b̃| > |v| and | |u+ b̃| − |v| | < Λ. (c) If |v| > 2Λ and k ∈ T0∩Td with d 6= 0, then |Nb(k)| ≥ Λ2 (|v|+ |u+b|) for all b 6∈ {0, d}. Furthermore we have |d| > |v| and | |u+ d| − |v| | < Λ. Proof. (a) By hypothesis, both factors in |Nb(k)| = |v + (u+ b)⊥| |v − (u+ b)⊥| are greater or equal to Λ. We now prove that at least one of the factors must also be greater or equal to 12(|v|+ |u+ b|). Suppose that |v| ≥ |u+ b|. Then, since (v + (u+ b)⊥) + (v − (u+ b)⊥) = 2v, 44 at least one of the factors must also be greater or equal to |v| = 1 2 (|v|+ |v|) ≥ 1 2 (|v|+ |u+ b|). Now suppose that |v| < |u+ b|. Then, since (v + (u+ b)⊥)− (v − (u+ b)⊥) = 2(u+ b)⊥, at least one of the factors must also be greater or equal to |u+ b| = 1 2 (|u+ b|+ |u+ b|) > 1 2 (|v|+ |u+ b|). All this together implies that |Nb(k)| ≥ Λ2 (|v|+ |u+ b|), which proves part (a). (b) By hypothesis ε < Λ/6 < 2Λ < |v|. Let k ∈ T0. Then, by (3.3.2), |N0(k)| ≤ ε(2|v|+ ε) < 3ε|v| < Λ2 |v|. (3.3.4) Thus we have either |u + v⊥| < Λ or |u − v⊥| < Λ (otherwise apply part (a) to get a contradiction). Suppose that |u+v⊥| < Λ. Then there is no b ∈ Γ#\{0} with |b+u+v⊥| < Λ and there is at most one b̃ ∈ Γ# \ {0} satisfying |b̃ + u − v⊥| < Λ. This inequality implies | |u+ b̃| − |v| | < Λ. Furthermore, for this b̃, |b̃| = |2v⊥ − (u+ v⊥) + (b̃+ u− v⊥)| > 2|v| − 2Λ > |v|, since −2Λ > −|v|. Now suppose that |u− v⊥| < Λ. Then there is no b ∈ Γ# \ {0} obeying |b + u − v⊥| < Λ and there is at most one b̃ ∈ Γ# \ {0} satisfying |b̃ + u + v⊥| < Λ. This inequality implies | |u+ b̃| − |v| | < Λ. Consequently, |b̃| = |2v⊥ − (v⊥ − u)− (v⊥ + b̃+ u)| > 2|v| − 2Λ > |v|. Finally observe that, if b 6∈ {0, b̃} then |b + u + v⊥| ≥ Λ and |b + u − v⊥| ≥ Λ. Hence, applying part (a) it follows that |Nb(k)| ≥ Λ2 (|v|+ |u+ b|). This proves part (b). (c) As in the proof of part (b), if k ∈ T0 ∩ Td then in addition to (3.3.4) we have |Nd(k)| ≤ ε(2|v|+ ε) < 3ε|v| < Λ2 |v|. Thus, applying part (b) we conclude that d must be the exceptional b̃ of part (b). The statement of part (c) follows then from part (b). This completes the proof of the proposition. 45 3.4 Motivation and main results Below we state our main results. The proofs come later divided in many steps. In [4], the authors introduced a class of Riemann surfaces of infinite genus that are “asymptotic to” a finite number of complex lines joined by infinite many handles. These surfaces are constructed by pasting together a compact submanifold of finite genus, plane domains, and handles. More precisely, these surfaces can be decomposed into Xcom ∪Xreg ∪Xhan, where Xcom is a compact submanifold with smooth boundary and finite genus, Xreg is a finite union of open “regular pieces”, and Xhan is an infinite union of closed “handles”. All these components satisfy a number of geometric/analytic hypotheses stated in [4] that specify the asymptotic holomorphic structure of the surface. The class of surfaces obtained in this way yields an extension of the classical theory of compact Riemann surfaces that has analogues of many theorems of the classical theory. The choice of geometric/analytic hypotheses was guided by two requirements. First, that the classical theory of compact Riemann surfaces could be developed in the new context. Secondly, that a number of interesting examples satisfy the hypotheses. In fact, it was proven in [4] that this new class of surfaces includes quite general hyperelliptic surfaces, heat curves (which are related to the Kadomcev-Petviashvilli equation), and Fermi curves with zero magnetic potential F(A = 0, V ). In order to verify the geometric/analytic hypotheses for F(0, V ) the authors proved two “asymptotic” theorems similar to the ones we prove below. This is the main step needed to verify the geometric/analytic hypotheses. In this thesis we extend their results to Fermi curves F(A, V ) with “small” magnetic potential A. We have followed their strategy of analysis. The main idea is to consider the eigenvalue- eigenvector problem for Hk(A, V ) for k ∈ C2 with large imaginary part as a perturbation of the problem for Hk(0, 0). When A is zero, they are able to prove asymptotic theorems for F(0, V ) for arbitrary large V . When A is not zero, however, new difficulties arise due to the presence of the term A · (i∇ − k) in the Hamiltonian Hk. When A is large, taking the imaginary part of k ∈ C2 arbitrarily large is not enough to control this term—it is not enough to make its contribution small and hence have the interacting Fermi curve as a perturbation of the free Fermi curve. (The term V in Hk is easily controlled by this 46 method.) However, as we shall see below, the proof can be implemented by assuming that A is small. Before we proceed we need to introduce some notation. For any ϕ ∈ L2(R2/Γ) define ϕ̂ : Γ# → C as ϕ̂(b) := (Fϕ)(b) := 1|Γ| ∫ R2/Γ ϕ(x) e−ib·x dx, where |Γ| := ∫ R2/Γ dx. Then, ϕ(x) = (F−1ϕ̂)(x) = ∑ b∈Γ# ϕ̂(b) eib·x and ‖ϕ‖L2(R2/Γ) = |Γ|1/2‖ϕ̂‖l2(Γ#). Let ρ be a positive constant and set Kρ := {k ∈ C2 | |v| ≤ ρ}. Consider the projection pr : C2 −→ C, (k1, k2) 7−→ k2, and define q := (i∇ ·A) +A2 + V. Finally, recall that Tν(b) = {k ∈ C2 | |Nb,ν(k)| = |v + (−1)ν(u+ b)⊥| < ε} and Tb := T1(b) ∪ T2(b). Clearly, the set Kρ is invariant under the action of Γ# and Kρ/Γ# is compact. Hence, the image of F̂(A, V ) ∩ Kρ under the exponential map E : F̂(A, V )→ F(A, V ) is compact in F(A, V ). It will essentially play the role of Xcom in the decomposition of F(A, V ). Our first theorem characterizes the regular piece Xreg. Theorem 3.4.1 (The regular piece). Let 0 < ε < Λ/6 and suppose that A1, A2 and V are functions in L2(R2/Γ) obeying ‖b2q̂(b)‖l1(Γ#) < ∞ and ‖(1 + b2)Â(b)‖l1(Γ#\{0}) < 2ε/63. Then there is a constant ρ = ρΛ,ε,q,A such that, for ν ∈ {1, 2}, the projection pr induces a biholomorphic map between ( F̂(A, V ) ∩ Tν(0) ) \ Kρ ∪ ⋃ b∈Γ#\{0} Tb 47 and its image in C. This image component contains{ z ∈ C ∣∣∣ 8|z| > ρ and |z + (−1)νθν(b)| > ε for all b ∈ Γ# \ {0}} and is contained in{ z ∈ C ∣∣∣∣∣ |z + (−1)νθν(b)| > 12 ( ε− ε 2 40Λ ) for all b ∈ Γ# \ {0} } , where θν(b) = 12((−1)νb2 + ib1). Furthermore, pr−1 : Image(pr) −→ Tν(0), y 7−→ (−β(1,0)2 − i(−1)νy − r(y), y), where β(1,0)2 is a constant given by (4.6.8) that depends only on ρ and A, |β(1,0)2 | < ε2 100Λ and |r(y)| ≤ ε 3 50Λ2 + C ρ , where C = CΛ,ε,q,A is a constant. Since Tb + c = Tb+c for all b, c ∈ Γ#, the complement of E (F̂(A, V ) ∩Kρ) in F(A, V ) is the disjoint union of E ((F̂(A, V ) ∩ T0) \(Kρ ∪ ⋃ b∈Γ# b2 6=0 Tb )) and ⋃ b∈Γ# b2 6=0 E (F̂(A, V ) ∩ T0 ∩ Tb). Basically, the first of the two sets will be the regular piece of F(A, V ), while the second set will be the handles. The map Φ parametrizing the regular part will be the composition of the exponential map E with the inverse of the map discussed in the above theorem. The detailed information about the handles comes from our second main theorem. Theorem 3.4.2 (The handles). Let 0 < ε < Λ/6 and suppose that A1, A2 and V are functions in L2(R2/Γ) with ‖b2q̂(b)‖l1(Γ#) <∞ and ‖(1+b2)Â(b)‖l1(Γ#\{0}) < 2ε/63. Then, for every sufficiently large constant ρ and for every d ∈ Γ# \ {0} with 2|d| > ρ, there are maps φd,1 : { (z1, z2) ∈ C2 ∣∣∣ |z1| ≤ ε2 and |z2| ≤ ε2} −→ T1(0) ∩ T2(d), φd,2 : { (z1, z2) ∈ C2 ∣∣∣ |z1| ≤ ε2 and |z2| ≤ ε} −→ T1(−d) ∩ T2(0), and a complex number td with |td| ≤ C|d|4 such that: 48 (i) For ν ∈ {1, 2} the domain of the map φd,ν is biholomorphic to its image, and the image contains{ k ∈ C2 ∣∣∣ |k1 + i(−1)νk2| ≤ ε8 and |k1 +(−1)ν+1d1− i(−1)ν(k2 +(−1)ν+1d2)| ≤ ε8}. Furthermore, Dφ̂d,ν = 1 2 1 1 −i(−1)ν i(−1)ν (I +O( 1|d|2 )) and φd,ν(0) = (iθν(d), (−1)ν+1θν(d)) +O ( ε 900 ) +O ( 1 ρ ) . (ii) φ−1d,1(T1(0) ∩ T2(d) ∩ F̂(A, V )) = { (z1, z2) ∈ C2 ∣∣∣ z1z2 = td, |z1| ≤ ε2 and |z2| ≤ ε2}, φ−1d,2(T1(−d) ∩ T2(0) ∩ F̂(A, V )) = { (z1, z2) ∈ C2 ∣∣∣ z1z2 = td, |z1| ≤ ε2 and |z2| ≤ ε2}. (iii) φd,1(z1, z2) = φd,2(z2, z1)− d. These are the main new results of this thesis. Using these theorems one should be able to verify that F(A, V ) satisfies the geometric/analytic hypotheses as was done in [4] for F(0, V ). In the next section we outline the strategy for proving these results. The proofs are given in the next chapter divided in many steps. We finally mention a small simplification (or modification) that we were able to make in the Theorem 3.4.1 (the regular piece). We shall not go into details here. We refer the reader to Chapter 5. First, recall an important property of F̂(A, V )—namely, gauge invariance. Briefly, gauge invariance implies that F̂(A, V ) = F̂(A+∇Ψ, V ), where Ψ is function on R2 (under suitable hypotheses), and∇Ψ is periodic with respect to Γ. In Chapter 5, by choosing a convenient gauge Ψ, we are able to indicate how to simplify the proof of Theorem 3.4.1 and improve some constants in it. After performing this gauge transformation “some terms vanish” and the analysis becomes simpler. We do not provide this “new” proof because it is essentially the same as the proof given below, up to some minor modifications. In fact, we believe that the simplifications introduced by the gauge will become clear after the reader gets familiar with the proof of the above results. 49 3.5 Strategy outline Below we briefly describe the general strategy of analysis used to prove our results. It was applied by Feldman, Knörrer and Trubowitz in [4, §16] for studying the case where A = 0. We implement it in detail in the subsequent sections for A 6= 0. Let us first introduce some notation and definitions. Observe that Hk(A, V )ϕ = ((i∇+A− k)2 + V )ϕ = ((i∇− k)2 +A · (i∇− k) + (i∇− k) ·A+A2 + V )ϕ = ((i∇− k)2 +A · (i∇− 2k) + (i∇ ·A) +A · i∇+A2 + V )ϕ = ((i∇− k)2 + 2A · (i∇− k) + (i∇ ·A) +A2 + V )ϕ, and write Hk(A, V ) = ∆k + h(k,A) + q(A, V ) with ∆k := (i∇− k)2, h(k,A) := 2A · (i∇− k) and q(A, V ) := (i∇ ·A) +A2 + V. For each finite subset G of Γ# set G′ := Γ# \G and C2G := C2 \ ⋃ b∈G′ Nb, L2G := span{eib·x | b ∈ G} and L2G′ := span{eib·x | b ∈ G′}. To simplify the notation write L2 in place of L2(R2/Γ), let I be the identity operator on L2, and let piG and piG′ be the orthogonal projections from L2 onto L2G and L 2 G′ , respectively. Then, L2 = L2G ⊕ L2G′ and I = piG + piG′ . For k ∈ C2G define the partial inverse (∆k)−1G on L2 as (∆k)−1G := piG + ∆ −1 k piG′ . Its matrix elements are ( (∆k)−1G ) b,c := 〈 eib·x |Γ|1/2 , (∆k) −1 G eic·x |Γ|1/2 〉 L2 = δb,c if c ∈ G, δb,c 1 Nc(k) if c 6∈ G, where b, c ∈ Γ#. 50 Here is the main idea: By definition, a point k is in F̂(A, V ) if Hk(A, V ) has a nontrivial kernel in L2. Hence, to study the part of the curve in the intersection of ∪d′∈GTd′ with C2 \ ∪b∈G′Tb for some finite subset G of Γ# (see Figure 3.6), it is natural to look for a nontrivial solution of (∆k + h+ q)(ψG + ψG′) = 0, where ψG ∈ L2G and ψG′ ∈ L2G′ . Equivalently, if we make the following (invertible) change of variables in L2, (ψG + ψG′) = (∆k)−1G (ϕG + ϕG′), where ϕG ∈ L2G and ϕG′ ∈ L2G′ , we may consider the equation (∆k + h+ q)ϕG + (I + (h+ q)∆−1k )ϕG′ = 0. (3.5.1) The projections of this equation onto L2G′ and L 2 G are, respectively, piG′(h+ q)ϕG + piG′(I + (h+ q)∆−1k )ϕG′ = 0, (3.5.2) piG(∆k + h+ q)ϕG + piG(h+ q)∆−1k ϕG′ = 0. (3.5.3) Now define RG′G′ on L2 as RG′G′ := piG′(I + (h+ q)∆−1k )piG′ . Observe that RG′G′ is the zero operator on L2G. Then, if RG′G′ has a bounded inverse on L2G′ , the equation (3.5.2) is equivalent to ϕG′ = −R−1G′G′piG′(h+ q)ϕG. Substituting this into (3.5.3) yields piG(∆k + h+ q − (h+ q)∆−1k R−1G′G′piG′(h+ q))ϕG = 0. This equation has a nontrivial solution if and only if the (finite) |G| × |G| determinant det [piG(∆k + h+ q − (h+ q)∆−1k R−1G′G′piG′(h+ q))piG ] = 0 or, equivalently, expressing all operators as matrices in the basis {|Γ|−1/2 eib·x | b ∈ Γ#}, det Nd′(k)δd′,d′′ + wd′,d′′ − ∑ b,c∈G′ wd′,b Nb(k) (R−1G′G′)b,cwc,d′′ d′,d′′∈G = 0, (3.5.4) 51 where wb,c := hb,c + q̂(b− c) = −2(c+ k) · Â(b− c) + q̂(b− c). Therefore, if RG′G′ has a bounded inverse on L2G′—which is in fact the case under suitable conditions—in the region under consideration we can study the Fermi curve in detail using the (local) defining equation (3.5.4). In order to implement this strategy we shall first derive a number of analytic estimates. ik1 k2 T2(0) T1(0) T0 ∩ (C2 \ ∪b∈Γ#\{0}Tb) ik1 k2 T2(0) T1(d) T1(0)T2(d) (T0 ∪ Td) ∩ (C2 \ ∪b∈Γ#\{0,d}Tb) Figure 3.6: Sketch of (∪d′∈GTd′) ∩ (C2 \ ∪b∈G′Tb) for G = {0} and G = {0, d}. 52 3.6 Notation and remarks We summarize here some notation and remarks that will be used henceforth. (i) For ν ∈ {1, 2} define the (complementary) index ν ′ as ν ′ := ν − (−1)ν . Observe that ν ′ = 2 if ν = 1 1 if ν = 2 and (−1)ν = −(−1)ν′ . (ii) Let f(x) and g(x) be multivariable functions and let p be a real number. The notation f(x) = O(|x|p) means that there is a constant C > 0 such that |f(x)| ≤ C|x|p for all x in a suitable domain. Similarly, the statement O(f(x)) = O(g(x)) is equivalent to say that there are constants C1 > 0 and C2 > 0 such that C1|f(x)| ≤ |g(x)| ≤ C2|f(x)|. (iii) Let z and w be vectors in C2. We denote the length of z by |z| := (z1z̄1 + z2z̄2)1/2 = (|z1|2 + |z2|2)1/2. Note that z · w := z1w1 + z2w2 is not an inner product on C2. However, we still have the property |z · w| = |z1w1 + z2w2| ≤ |z1| |w1|+ |z2| |w2| = (|z1|, |z2|) · (|w1|, |w2|) ≤ |z| |w|. Here we have used the Schwarz inequality (for the inner product on R2). 53 (iv) Let T be a linear operator from L2C to L 2 B with B,C ⊂ Γ#. We denote its matrix elements in the basis {|Γ|−1/2 eib·x | b ∈ Γ#} by Tb,c := 〈 eib·x |Γ|1/2 , T eic·x |Γ|1/2 〉 L2 , where b ∈ B and c ∈ C. The operator T represented as a matrix [Tb,c] acts on l2(Γ#). (v) Consider a linear operator T : X → Y . The operator norm of T is defined as ‖T‖ := sup x∈X ‖Tx‖Y ‖x‖X . (vi) In general, we denote by CX1,...,Xn a constant that depends only on X1, . . . , Xn. We may use the same symbol CX1,...,Xn to denote different constants that change from line to line in our calculations without further notice. (vii) The following notation will be used whenever we consider vector-valued quantities. Let X be a Banach space and let A = (A1, . . . , Ad) ∈ X d and B = (B1, . . . , Bd) ∈ X d. Then, ‖A‖X := (‖A1‖2X + · · ·+ ‖Ad‖2X )1/2 and A ·B := A1B1 + · · ·+AdBd. (viii) Finally, recall the Neumann series for bounded linear operators, (I − T )−1 = ∞∑ j=0 T j , and the the geometric series for complex numbers, 1 1− z = ∞∑ j=0 zj , which are convergent if ‖T‖ < 1 and |z| < 1. 54 Chapter 4 Weak magnetic potential 4.1 Invertibility of RG′G′ A simple strategy for inverting RG′G′ is the following. Since this operator has the form I + T , it is easily invertible if ‖T‖ < 1. In this section we manage to get this bound and prove that for a suitable choice of G ⊂ Γ#, large |v|, and weak magnetic potential, the operator RG′G′ is invertible on L2G′ . In general, for any B,C ⊂ Γ# (C such that ∆−1k piC exists) define the operator RBC as RBC :=piB(I + (h+ q)∆−1k )piC =piBpiC + piB q∆−1k piC + piB(2A · i∇)∆−1k piC − piB(2k ·A)∆−1k piC . (4.1.1) Its matrix elements are (RBC)b,c = δb,c + q̂(b− c) Nc(k) − 2c · Â(b− c) Nc(k) − 2k · Â(b− c) Nc(k) , (4.1.2) where b ∈ B and c ∈ C. We shall first estimate the norm of the last three terms on the right hand side of (4.1.1). We begin with the following proposition. Proposition 4.1.1. Let k ∈ C2 and let B,C ⊂ Γ# with C ⊂ {b ∈ Γ# | Nb(k) 6= 0}. Then, ‖piB q∆−1k piC‖ ≤ ‖q̂‖l1 sup c∈C 1 |Nc(k)| , ‖piB(A · i∇)∆−1k piC‖ ≤ ‖Â‖l1 sup c∈C |c| |Nc(k)| , ‖piB(k ·A)∆−1k piC‖ ≤ ‖Â‖l1 |k| sup c∈C 1 |Nc(k)| . 55 To prove this proposition we apply the following inequality, which we prove later in §4.9. Proposition 4.1.2. Consider a linear operator T : L2C → L2B with matrix elements Tb,c. Then, ‖T‖ ≤ max { sup c∈C ∑ b∈B |Tb,c|, sup b∈B ∑ c∈C |Tb,c| } . Proof of Proposition 4.1.1. Write T1 := piB q∆−1k piC . Then, in view of (4.1.1) and (4.1.2), sup c∈C ∑ b∈B |(T1)b,c| ≤ sup c∈C ∑ b∈B |q̂(b− c)| |Nc(k)| ≤ supc∈C 1 |Nc(k)|‖q̂‖l1 , sup b∈B ∑ c∈C |(T1)b,c| ≤ sup b∈B ∑ c∈C |q̂(b− c)| |Nc(k)| ≤ supc∈C 1 |Nc(k)|‖q̂‖l1 . By Proposition 4.1.2, these estimates imply the first inequality. Now, let T2 := piB(A · i∇)∆−1k piC and T3 := piB(k · A)∆−1k piC . Similarly as above, the second and third inequalities follow from the estimates sup c∈C ∑ b∈B |(T2)b,c| ≤ sup c∈C ∑ b∈B |c| |Â(b− c)| |Nc(k)| ≤ supc∈C |c| |Nc(k)|‖Â‖l1 , sup b∈B ∑ c∈C |(T2)b,c| ≤ sup b∈B ∑ c∈C |c| |Â(b− c)| |Nc(k)| ≤ supc∈C |c| |Nc(k)|‖Â‖l1 , and sup c∈C ∑ b∈B |(T3)b,c| ≤ sup c∈C ∑ b∈B |k| |Â(b− c)| |Nc(k)| ≤ supc∈C 1 |Nc(k)| |k| ‖Â‖l1 , sup b∈B ∑ c∈C |(T3)b,c| ≤ sup b∈B ∑ c∈C |k| |Â(b− c)| |Nc(k)| ≤ supc∈C 1 |Nc(k)| |k| ‖Â‖l1 . This proves the proposition. The key estimate for the existence of R−1G′G′ is given below. Proposition 4.1.3 (Estimate of ‖RSS − piS‖). Let k ∈ C2 with |u| ≤ 2|v| and |v| > 2Λ. Suppose that S ⊂ {b ∈ Γ# | |Nb(k)| ≥ ε|v|}. Then, ‖RSS − piS‖ ≤ ‖q̂‖l1 1 ε|v| + 14 ε ‖Â‖l1 . (4.1.3) If A = 0 the right hand side of (4.1.3) can be made arbitrarily small for any q(0, V ) = V by taking |v| sufficiently large. If A 6= 0, however, we need to take ‖Â‖l1 small to make that quantity less than 1. The term 14ε ‖Â‖l1 in (4.1.3) comes from the estimate we have for ‖piG′ h∆−1k piG′‖. 56 Proof. By hypothesis, for all b ∈ S, 1 |Nb(k)| ≤ 1 ε|v| . (4.1.4) We now show that, for all b ∈ S, |b| |Nb(k)| ≤ 4 ε . (4.1.5) First suppose that |b| ≤ 4|v|. Then, |b| |Nb(k)| ≤ 4|v| ε|v| = 4 ε . Now suppose that |b| ≥ 4|v|. Again, by hypothesis we have |u| ≤ 2|v| and |v| > 2Λ > Λ/6 > ε. Hence, |v ± (u+ b)⊥| ≥ |b| − |u| − |v| ≥ |b| − 3|v| ≥ |b| − 3 4 |b| = |b| 4 . Consequently, |b| |Nb(k)| = |b| |v + (u+ b)⊥| |v − (u+ b)⊥| ≤ |b| 4 |b| 4 |b| = 16 |b| ≤ 4 |v| ≤ 4 ε . This proves (4.1.5). The expression for RSS − piS is given by (4.1.1). Observe that |k| = |u+ iv| ≤ |u|+ |v| ≤ 3|v|. Then, applying Proposition 4.1.1 and using (4.1.4) and (4.1.5) we obtain ‖RSS − piS‖ ≤ (6|v| ‖Â‖l1 + ‖q̂‖l1) sup b∈S 1 |Nc(k)| + 2‖Â‖l1 supb∈S |c| |Nc(k)| ≤ (6|v| ‖Â‖l1 + ‖q̂‖l1) 1 ε|v| + 8 ε ‖Â‖l1 = ‖q̂‖l1 1 ε|v| + 14 ε ‖Â‖l1 . This is the desired inequality. From the last proposition it follows easily that RSS has a bounded inverse for large |v| and weak magnetic potential. Lemma 4.1.4 (Invertibility of RSS). Let k ∈ C2, |u| ≤ 2|v|, |v| > max { 2Λ, ‖q̂‖l1 2 ε } , ‖q̂‖l1 <∞ and ‖Â‖l1 < 2 63 ε. 57 Suppose that S ⊂ {b ∈ Γ# | |Nb(k)| ≥ ε|v|}. Then the operator RSS has a bounded inverse with ‖RSS − piS‖ < ‖q̂‖l1 1 ε|v| + ‖Â‖l1 14 ε < 17 18 and ‖R−1SS − piS‖ < 18‖RSS − piS‖. Proof. Write RSS = piS + T with T = RSS − piS . Then, by Proposition 4.1.3, ‖T‖ = ‖RSS − piS‖ ≤ ‖q̂‖l1 1 ε|v| + ‖Â‖l1 14 ε < 1 2 + 4 9 = 17 18 < 1. Hence, the Neumann series for R−1SS = (piS + T ) −1 converges (and is a bounded operator). Furthermore, ‖R−1SS − piS‖ = ‖(piS + T )−1 − piS‖ = ‖(piS + T )−1 − (piS + T )−1(piS + T )‖ = ‖(piS + T )−1T‖ ≤ (1− ‖T‖)−1‖T‖ < 18‖RSS − piS‖, as was to be shown. Lemma 4.1.4 says that if G is such that G′ ⊂ {b ∈ Γ# | |Nb(k)| ≥ ε|v|}, the operator RG′G′ has a bounded inverse on L2G′ for |u| ≤ 2|v|, large |v|, and weak magnetic potential. We are now able to write (local) defining equations for F̂(A, V ) under such conditions. 4.2 Local defining equations In this section we derive (local) defining equations for the Fermi curve. We begin with a simple proposition. Proposition 4.2.1. Suppose either (i) or (ii) or (iii) where: (i) G = {0} and k ∈ T0 \ ∪b∈Γ#\{0}Tb; (ii) G = {0, d} and k ∈ T0 ∩ Td; (iii) G = ∅ and k ∈ C2 \ ∪b∈Γ#Tb. Then G′ = Γ# \G = {b ∈ Γ# | |Nb(k)| ≥ ε|v|}. 58 Proof. The proposition follows easily if we observe that G′ = Γ# \G and recall from (3.3.1) that k 6∈ Tb =⇒ |Nb(k)| ≥ ε|v|. We now need some notation. Let B be a fundamental cell for Γ# ⊂ R2 (see [12, p 310]). Then any vector u ∈ R2 can be written as u = ξ + u, for some ξ ∈ Γ# and u ∈ B. Define α := sup{|u| | u ∈ B}, R := max { α, 2Λ, ‖q̂‖l1 2 ε } , KR := {k ∈ C2 | |v| ≤ R}. We first show that in C2 \ KR the Fermi curve is contained in the union of ε-tubes about the free Fermi curve. Proposition 4.2.2 (F̂(A, V ) \ KR is contained in the union of ε-tubes). F̂(A, V ) \ KR ⊂ ⋃ b∈Γ# Tb. Proof. Recall that F̂(A, V ) is invariant under the action of Γ#, that is, that k ∈ F̂(A, V ) if and only if k + ξ ∈ F̂(A, V ) for all ξ ∈ Γ#. Hence, given any k ∈ F̂(A, V ), we can always perform a change of coordinates k → k−ξ for some suitable ξ ∈ Γ# so that k−ξ ∈ F̂(A, V ) and Re(k − ξ) ∈ B. Here (Re k) ∈ R2 denotes the real part of k ∈ C2. Thus, without loss of generality we may assume that (Re k) ∈ B. We now prove that any point outside the region KR and outside the union of ε-tubes does not belong to F̂(A, V ). Suppose that k ∈ C2 \ (KR ∪ ⋃ b∈Γ# Tb) and recall that k is in F̂(A, V ) if and only if (3.5.1) has a nontrivial solution. If we choose G = ∅ then G′ = Γ# and this equation reads RG′G′ϕG′ = 0. By Proposition 4.2.1(iii) we have G′ = Γ# = {b ∈ Γ# | |Nb(k)| ≥ ε|v|}. Furthermore, since u ∈ B and |v| > R ≥ α, it follows that |u| ≤ α < |v| < 2|v|. Consequently, the operator RG′G′ has a bounded inverse by Lemma 4.1.4. Thus, the only solution of the above equation is ϕG′ = 0. That is, there is no nontrivial solution of this equation and therefore k 6∈ F̂(A, V ). 59 We are left to study the Fermi curve inside the ε-tubes. There are two types of regions to consider: intersections and non-intersections of tubes. To study non-intersections we choose G = {0} and consider the region (T0 \ ∪b∈Γ#\{0}Tb) \ KR. For intersections we take G = {0, d} for some d ∈ Γ# \{0} and consider (T0∩Td)\KR (see Figure 4.1). Observe that, since the tubes Tb have the following translational property, Tb + c = Tb+c for all b, c ∈ Γ#, and the curve F̂(A, V ) is invariant under the action of Γ#, there is no loss of generality in considering only the two regions above. Any other part of the curve can be reached by translation. KR KR Figure 4.1: Sketch of ( T0 \ ∪b∈Γ#\{0}Tb ) \ KR (left) and (T0 ∩ Td) \ KR (right). Recall that G′ = Γ# \G, and for d′, d′′ ∈ G and i, j ∈ {1, 2} set Bd ′d′′ ij (k;G) := −4 ∑ b,c∈G′ Âi(d′ − b) Nb(k) (R−1G′G′)b,c Âj(c− d′′), Cd ′d′′ i (k;G) := −2Âi(d′ − d′′) + 2 ∑ b,c∈G′ q̂(d′ − b)− 2b · Â(d′ − b) Nb(k) (R−1G′G′)b,cÂi(c− d′′) + 2 ∑ b,c∈G′ Âi(d′ − b) Nb(k) (R−1G′G′)b,c(q̂(c− d′′)− 2d′′ · Â(c− d′′)), Cd ′d′′ 0 (k;G) := q̂(d ′ − d′′)− 2d′′ · Â(d′ − d′′) − ∑ b,c∈G′ q̂(d′ − b)− 2b · Â(d′ − b) Nb(k) (R−1G′G′)b,c(q̂(c− d′′)− 2d′′ · Â(c− d′′)). (4.2.1) 60 Then, Dd′,d′′(k;G) := wd′,d′′ − ∑ b,c∈G′ wd′,b Nb(k) (R−1G′G′)b,cwc,d′′ = q̂(d′ − d′′)− 2(d′′ + k) · Â(d′ − d′′) − ∑ b,c∈G′ (q̂(d′ − b)− 2(b+ k) · Â(d′ − b))(R −1 G′G′)b,c Nb(k) (q̂(c− d′′)− 2(d′′ + k) · Â(c− d′′)) = Bd ′d′′ 11 k 2 1 +B d′d′′ 22 k 2 2 + (B d′d′′ 12 +B d′d′′ 21 )k1k2 + C d′d′′ 1 k1 + C d′d′′ 2 k2 + C d′d′′ 0 . Furthermore, we shall shortly prove the following property. Proposition 4.2.3. For d′, d′′ ∈ G and i, j ∈ {1, 2}, the functions Bd′d′′ij , Cd ′d′′ i , C d′d′′ 0 (and consequently Dd′,d′′) are analytic on (T0 \∪b∈Γ#\{0}Tb) \KR and (T0 ∩Td) \KR for G = {0} and G = {0, d}, respectively. Using the above functions we can write (local) defining equations for the Fermi curve. Lemma 4.2.4 (Local defining equations for F̂(A, V )). (i) Let G = {0} and k ∈ (T0 \ ∪b∈Γ#\{0}Tb) \ KR. Then k ∈ F̂(A, V ) if and only if N0(k) +D0,0(k) = 0. (ii) Let G = {0, d} and k ∈ (T0 ∩ Td) \ KR. Then k ∈ F̂(A, V ) if and only if (N0(k) +D0,0(k))(Nd(k) +Dd,d(k))−D0,d(k)Dd,0(k) = 0. We now prove this lemma and then Proposition 4.2.3. The proof of this lemma is easy once we have that RG′G′ is invertible. Proof. (i) First, by Proposition 4.2.1(i) we have G′ = Γ# \ {0} = {b ∈ Γ# | |Nb(k)| ≥ ε|v|}. Furthermore, since k ∈ T0, we have either |v − u⊥| < ε or |v + u⊥| < ε. In either case this implies |u| < ε+ |v| < 2Λ + |v| < 2|v|. (4.2.2) Hence, the operator RG′G′ has a bounded inverse by Lemma 4.1.4. Thus, in the region under consideration F̂(A, V ) is given by (3.5.4): 0 = N0(k) + w0,0 − ∑ b,c∈G′ w0,b Nb(k) (R−1G′G′)b,cwc,0 = N0(k) +D0,0(k). This proves part (i). 61 (ii) Similarly, by Proposition 4.2.1(ii), G′ = Γ# \ {0, d} = {b ∈ Γ#| |Nb(k)| ≥ ε|v|}. Furthermore, since k ∈ T0 ∩Td ⊂ T0, similarly as above we obtain (4.2.2). Thus, by Lemma 4.1.4 the operator RG′G′ has a bounded inverse. Hence, in the region under consideration F̂(A, V ) is given by (3.5.4): (N0(k) +D0,0(k))(Nd(k) +Dd,d(k))−D0,d(k)Dd,0(k) = 0. This proves part (ii) and completes the proof of the lemma. As promised, here is the (sketchy) proof of Proposition 4.2.3. Proof of Proposition 4.2.3. It suffices to show that Bd ′d′′ ij , C d′d′′ i and C d′d′′ 0 are analytic func- tions. This property follows from the fact that all the series involved in the definition of these functions are uniformly convergent sums of analytic functions (see [16, Theorem 4.1]). The argument is similar for all cases. We give only a sketch of the proof. First observe that, in view of Lemma 4.1.4, for each b, c ∈ G′, |(R−1G′G′)b,c| = ∣∣((piG′ − (piG′ −RG′G′))−1)b,c∣∣ = ∣∣∣∣∣ ∞∑ j=0 ( (piG′ −RG′G′)j ) b,c ∣∣∣∣∣ ≤ ∞∑ j=0 ‖piG′ −RG′G′‖j < ∞∑ j=0 ( 17 18 )j = 18. Hence, the above sum converges uniformly by the Weierstrass M-test. Since (piG′−RG′G′)b,c is an analytic function of k, so is (R−1G′G′)b,c. Now observe that Bd ′d′′ ij , C d′d′′ i and C d′d′′ 0 are given by sums of the form const + ∑ b,c∈G′ f(b, d′) Nb(k) (R−1G′G′)b,c g(c, d ′′), where f and g are known functions. Furthermore, all the terms in these series are analytic functions, and the sum converges uniformly because of the uniform bounds |b| |Nb(k)| ≤ 4 ε , 1 |Nb(k)| ≤ 1 ε|v| < 1 εR and |(R−1G′G′)b,c| ≤ ‖R−1G′G′‖ ≤ 18 for all b, c ∈ G′, and because f(·, d′) and g(·, d′′) are in l1(Γ#) in all cases by hypothesis. Consequently, all the limits Bd ′d′′ ij , C d′d′′ i and C d′d′′ 0 are analytic functions. 62 To study in detail the defining equations above we shall estimate the asymptotic be- haviour of the functions Bd ′d′′ ij , C d′d′′ i , C d′d′′ 0 and Dd′,d′′ for large |v|. (We sometimes refer to these functions as coefficients.) Since all these functions have a similar form it is convenient to prove these estimates in a general setting and specialize them later. This is the contents of §4.4 and §4.5. We next introduce a change of variables in C2 that will be useful for proving these bounds. 4.3 Change of coordinates The following change of coordinates in C2 will be useful for our analysis. For ν ∈ {1, 2} and d′, d′′ ∈ G define the functions wν,d′ , zν,d′ : C2 → C as wν,d′(k) := k1 + d′1 + i(−1)ν(k2 + d′2), zν,d′(k) := k1 + d′1 − i(−1)ν(k2 + d′2). (4.3.1) Observe that, the transformation (k1, k2) 7→ (wν,d′ , zν,d′) is just a translation composed with a rotation. Furthermore, if k ∈ Tν(d′)\KR then |wν,d′(k)| is “small” and |zν,d′(k)| is “large”. Indeed, |wν,d′(k)| = |Nd′,ν(k)| < ε and |zν,d′(k)| = |Nd′,ν′(k)| ≥ |v| > R. Define also Jd ′d′′ ν := 1 4(B d′d′′ 11 −Bd ′d′′ 22 + i(−1)ν(Bd ′d′′ 12 +B d′d′′ 21 )), Kd ′d′′ := 12(B d′d′′ 11 +B d′d′′ 22 ), Ld ′d′′ ν := −d′1Bd ′d′′ 11 − i(−1)νd′2Bd ′d′′ 22 − 12(d′2 + i(−1)νd′1)(Bd ′d′′ 12 +B d′d′′ 21 ) + 12(C d′d′′ 1 + i(−1)νCd ′d′′ 2 ), Md ′d′′ := d′21 B d′d′′ 11 + d ′2 2 B d′d′′ 22 + d ′ 1d ′ 2(B d′d′′ 12 +B d′d′′ 21 )− d′1Cd ′d′′ 1 − d′2Cd ′d′′ 2 + C d′d′′ 0 , where Jd ′d′′ ν , K d′d′′ , Ld ′d′′ ν and M d′d′′ are functions of k ∈ C2 that also depend on the choice of G ⊂ Γ#. Using these functions we can express Nd′(k)+Dd′,d′(k) and Dd′,d′′(k) as follows. Proposition 4.3.1. Let ν ∈ {1, 2} and let d′, d′′ ∈ G. Then, Nd′ +Dd′,d′ = Jd ′d′ ν′ w 2 ν,d′ + J d′d′ ν z 2 ν,d′ + (1 +K d′d′)wν,d′zν,d′ + Ld ′d′ ν′ wν,d′ + L d′d′ ν zν,d′ +M d′d′ 63 and Dd′,d′′ = Jd ′d′′ ν′ w 2 ν,d′ + J d′d′′ ν z 2 ν,d′ +K d′d′′wν,d′zν,d′ + Ld ′d′′ ν′ wν,d′ + L d′d′′ ν zν,d′ +M d′d′′ . Furthermore, Jd ′d′′ ν (k) = − ∑ b,c∈G′ (1,−i(−1)ν) · Â(d′ − b) Nb(k) (R−1G′G′)b,c (1,−i(−1)ν) · Â(c− d′′), Kd ′d′′(k) = −2 ∑ b,c∈G′ Â(d′ − b) · Â(c− d′′) Nb(k) (R−1G′G′)b,c, Ld ′d′′ ν (k) = ∑ b,c∈G′ q̂(d′ − b) + 2(d′ − b) · Â(d′ − b) Nb(k) (R−1G′G′)b,c(1, i(−1)ν) · Â(c− d′′) + ∑ b,c∈G′ (1, i(−1)ν) · Â(d′ − b) Nb(k) (R−1G′G′)b,c (q̂(c− d′′) + 2(d′ − d′′) · Â(c− d′′)) − (1, i(−1)ν) · Â(d′ − d′′), and Md ′d′′(k) = − ∑ b,c∈G′ q̂(d′ − b) + 2(d′ − b) · Â(d′ − b) Nb(k) (R−1G′G′)b,c q̂(c− d′′) + q̂(d′ − d′′) + 2(d′ − d′′) · Â(d′ − d′′). Proof. To simplify the notation write w = wν,d′ , z = zν,d′ , Bij = Bd ′d′′ ij and Ci = C d′d′′ i . First observe that, in view of (4.3.1), Nd′ = (k1 + d′1 + i(−1)ν(k2 + d′2))(k1 + d′1 − i(−1)ν(k2 + d′2)) = wz. Furthermore, k1 = 12(w + z)− d′1, k2 = (−1)ν 2i (w − z)− d′2, k21 = 1 4(w 2 + z2) + 12wz − d′1(w + z) + d′21 , k22 = −14(w2 + z2) + 12wz + i(−1)νd′2(w − z) + d′22 , k1k2 = i(−1)ν 4 (z 2 − w2)− 12(d′2 − i(−1)νd′1)w − 12(d′2 + i(−1)νd′1) + d′1d′2. 64 Hence, Dd′,d′′ = B11k21 +B22k 2 2 + (B12 +B21)k1k2 + C1k1 + C2k2 + C0 = 14(B11 −B22 − i(−1)ν(B12 +B21))w2 + 14(B11 −B22 + i(−1)ν(B12 +B21))z2 + (− d′1B11 + i(−1)νd′2B22 − 12(d′2 − i(−1)νd′1)(B12 +B21) + 12(C1 − i(−1)νC2))w + (− d′1B11 + i(−1)νd′2B22 − 12(d′2 + i(−1)νd′1)(B12 +B21) + 12(C1 + i(−1)νC2))z + d′21 B11 + d ′2 2 B22 + d ′ 1d ′ 2(B12 +B21)− d′1C1 − d′2C2 + C0 + 12(B11 +B22)wz = Jd ′d′′ ν′ w 2 + Jd ′d′′ ν z 2 +Kd ′d′′wz + Ld ′d′′ ν′ w + L d′d′′ ν z +M d′d′′ . This proves the first claim. Consequently, Nd′ +Dd′,d′ = Jd ′d′ ν′ w 2 + Jd ′d′ ν z 2 + (1 +Kd ′d′)wz + Ld ′d′ ν′ w + L d′d′ ν z +M d′d′ , which proves the second claim. Now, again to simplify the notation write fg = ∑ b,c∈G′ f̂(b, d′) Nb(k) (R−1G′G′)b,c ĝ(c, d ′′), that is, to represent sums of this form suppress the summation and the other factors. Note that fg 6= gf according to this notation. Then, substituting (4.2.1) into the definitions of Jd ′d′′ ν , K d′d′′ , Ld ′d′′ ν and M d′d′′ we obtain Jd ′,d′′ ν = 1 4(B11 −B22 + i(−1)ν(B12 +B21)) = −A1A1 +A2A2 − i(−1)ν(A1A2 +A2A1) = (A1 − i(−1)νA2)(−A1 + i(−1)νA2) = −((1,−i(−1)ν) ·A) ((1,−i(−1)ν) ·A) = − ∑ b,c∈G′ (1,−i(−1)ν) · Â(d′ − b) Nb(k) (R−1G′G′)b,c (1,−i(−1)ν) · Â(c− d′′) and Kd ′,d′′ = 12(B11 +B22) = −2(A1A1 +A2A2) = −2 ∑ b,c∈G′ Â(d′ − b) · Â(c− d′′) Nb(k) (R−1G′G′)b,c and 65 Ld ′d′′ ν = −d′1B11 − i(−1)νd′2B22 − 12(d′2 + i(−1)νd′1)(B12 +B21) + 12(C1 + i(−1)νC2) = 4d′1A1A1 + 4i(−1)νd′2A2A2 + 2d′2(A1A2 +A2A1) + 2i(−1)νd′1(A1A2 +A2A1) + (q − 2b ·A)A1 +A1(q − 2d′′ ·A) + i(−1)ν(q − 2b ·A)A2 + i(−1)νA2(q − 2d′′ ·A) − (Â1(d′ − d′′) + i(−1)νÂ2(d′ − d′′)) = 2(2d′1 − d′′1)A1A1 + 2i(−1)ν(2d′2 − d′′2)A2A2 + 2(d′2 − d′′2 + i(−1)νd′1)A1A2 + 2(d′2 + i(−1)νd′1 − i(−1)νd′′1)A2A1 + (q − 2b ·A)A1 +A1q + i(−1)ν(q − 2b ·A)A2 + i(−1)νA2q − (1, i(−1)ν) · Â(d′ − d′′) = 2(d′1A1 + d ′ 2A2)(A1 + i(−1)νA2) + 2(1, i(−1)ν) ·A ((d′ − d′′) ·A) + (q − 2b ·A)(A1 + i(−1)νA2) + (A1 + i(−1)νA2)q − (1, i(−1)ν) · Â(d′ − d′′) = (q + 2(d− b) ·A)(1, i(−1)ν) ·A+ (1, i(−1)ν) ·A (q + 2(d′ − d′′) ·A) − (1, i(−1)ν) · Â(d′ − d′′) so that Ld ′d′′ ν = ∑ b,c∈G′ q̂(d′ − b) + 2(d′ − b) · Â(d′ − b) Nb(k) (R−1G′G′)b,c(1, i(−1)ν) · Â(c− d′′) + ∑ b,c∈G′ (1, i(−1)ν) · Â(d′ − b) Nb(k) (R−1G′G′)b,c (q̂(c− d′′) + 2(d′ − d′′) · Â(c− d′′)) − (1, i(−1)ν) · Â(d′ − d′′), Md ′d′′ = d′21 B11 + d ′2 2 B22 + d ′ 1d ′ 2(B12 +B21)− d′1C1 − d′2C2 + C0 = −4(d′21 A1A1 + d′22 A2A2 + d′1d′2A1A2 + d′1d′2A2A1) + 2d′ · Â(d′ − d′′) − 2d′1(q − 2b ·A)A1 − 2d′1A1(q − 2d′ ·A)− 2d′2(q − 2b ·A)A2 − 2d′2A2(q − 2d′ ·A) − (q − 2b ·A)(q − 2d′ ·A) + q̂(d′ − d′′)− 2d′′ · Â(d′ − d′′) = −4(d′ ·A)(d′ ·A)− 2(q − 2b ·A)(d′ ·A)− 2(d′ ·A)(q − 2d′ ·A) − (q − 2b ·A)(q − 2d′ ·A) + q̂(d′ − d′′) + 2(d′ − d′′) · Â(d′ − d′′) = −qq − 2(d− b) ·Aq + q̂(d′ − d′′) + 2(d′ − d′′) · Â(d′ − d′′) = − ∑ b,c∈G′ q̂(d′ − b) + 2(d′ − b) · Â(d′ − b) Nb(k) (R−1G′G′)b,c q̂(c− d′′) + q̂(d′ − d′′) + 2(d′ − d′′) · Â(d′ − d′′). We next use this change of variables for deriving asymptotics for certain functions. 66 4.4 Asymptotics for the coefficients Let f and g be functions on Γ# and for k ∈ C2 and d′, d′′ ∈ G set Φd′,d′′(k;G) := ∑ b,c∈G′ f(d′ − b) Nb(k) (R−1G′G′)b,c g(c− d′′). (4.4.1) In this section we study the asymptotic behaviour of the function Φd′,d′′(k) for k in the union of ε-tubes with large |v|. We first give all the statements and then the proofs. Reset the constant R as R := max { 1, α, 2Λ, 140‖Â‖l1 , ‖(1 + b2)q̂(b)‖l1 4 ε } , (4.4.2) and make the following hypothesis. Hypothesis 4.4.1. ‖b2q̂(b)‖l1 <∞ and ‖(1 + b2)Â(b)‖l1 < 2 63 ε. Our first lemma provides and expansion for Φd′,d′(k) “in powers of 1/|zν,d′(k)|”. Lemma 4.4.1 (Asymptotics for Φd′,d′(k)). Under Hypothesis 4.4.1, let ν ∈ {1, 2} and let f and g be functions on Γ# with ‖b2f(b)‖l1 <∞ and ‖b2g(b)‖l1 <∞. Suppose either (i) or (ii) where: (i) G = {0} and k ∈ (Tν(0) \ ∪b∈G′Tb) \ KR; (ii) G = {0, d} and k ∈ (Tν(0) ∩ Tν′(d)) \ KR. Then, for (µ, d′) = (ν, 0) if (i) or (µ, d′) ∈ {(ν, 0), (ν ′, d)} if (ii), Φd′,d′(k) = α (1) µ,d′(k) + α (2) µ,d′(k) + α (3) µ,d′(k), where for 1 ≤ j ≤ 2, |α(j)µ,d′(k)| ≤ Cj (2|zµ,d′(k)| −R)j and |α (3) µ,d′(k)| ≤ C3 |zµ,d′(k)|R2 , where Cj = Cj;Λ,A,q,f,g and C3 = C3;ε,Λ,A,q,f,g are constants. Furthermore, the functions α (j) µ,d′(k) are given by (4.4.25) and (4.4.28) and are analytic in the region under consideration. 67 Below we have more information about the function α(1)µ,d′(k). Lemma 4.4.2 (Asymptotics for α(1)µ,d′(k)). Consider the same hypotheses of Lemma 4.4.1. Then, for (µ, d′) = (ν, 0) if (i) or (µ, d′) ∈ {(ν, 0), (ν ′, d)} if (ii), zµ,d′(k)α (1) µ,d′(k) = α (1,0) µ,d′ + α (1,1) µ,d′ (w(k)) + α (1,2) µ,d′ (k) + α (1,3) µ,d′ (k), where α(1,0)µ,d′ is a constant given by (4.4.39), and the remaining functions α (1,j) µ,d′ are given by (4.4.38). Furthermore, for 0 ≤ j ≤ 2, |α(1,j)µ,d′ | ≤ Cj and |α(1,3)µ,d′ | ≤ C3 2|zµ,d′(k)| −R, where Cj = Cj;Λ,A,f,g and C3 = C3;Λ,A,f,g are constants given by (4.4.40). The next lemma estimates the decay of Φd′,d′′(k) with respect to zν′,d(k) for d′ 6= d′′. Lemma 4.4.3 (Decay of Φd′,d′′(k) for d′ 6= d′′). Under Hypothesis 4.4.1, let ν ∈ {1, 2} and let f and g be functions on Γ# with ‖b2f(b)‖l1 < ∞ and ‖b2g(b)‖l1 < ∞. Suppose further that G = {0, d} and k ∈ (Tν(0) ∩ Tν′(d)) \ KR. Then, for d′, d′′ ∈ G with d′ 6= d′′, |Φd′,d′′(k)| ≤ CΓ#,ε,f,g |zν′,d(k)|3−10−1 , where CΓ#,ε,f,g is a constant. The above lemmas are the main statements of this section. Before we move to the proofs we give one more proposition that provides relations between the quantities |v|, |k2|, |zν,d′(k)| and |d| for k in the ε-tubes with large |v|. Proposition 4.4.4. For ν ∈ {1, 2} we have: (i) Let k ∈ Tν(0) \ KR. Then, 1 |zν,0(k)| ≤ 1 |v| ≤ 3 |zν,0(k)| and 1 4|v| ≤ 1 |k2| ≤ 8 |v| . (ii) Let k ∈ (Tν(0) ∩ Tν′(d)) \ KR. Then, 1 |zν,0(k)| ≤ 1 |v| ≤ 3 |zν,0(k)| , 1 |zν′,d(k)| ≤ 1 |v| ≤ 3 |zν′,d(k)| and 1 2|zν′,d(k)| ≤ 1 |d| ≤ 2 |zν′,d(k)| . This proposition will be used several times henceforth. We next prove all the above statements. 68 Proof of Proposition 4.4.4 Proof of Proposition 4.4.4. We first derive a more general inequality and then we prove parts (i) and (ii). First observe that, if k ∈ Tµ(d′) \ KR then |v + (−1)µ(u+ d′)⊥| = |Nd′,µ(k)| < ε < |v|. Hence, |v| ≤ |2v − (v + (−1)µ(u+ d′)⊥)| ≤ 3|v|. But |2v − (v + (−1)µ(u+ d′)⊥)| = |v − (−1)µ(u+ d′)⊥| = |k1 + d′1 − i(−1)µ(k2 + d′2)| = |zµ,d′(k)|. Thus, |v| ≤ |zµ,d′(k)| ≤ 3|v|. Therefore, 1 |zµ,d′(k)| ≤ 1 |v| ≤ 3 |zµ,d′(k)| . (4.4.3) (i) The first inequality of part (i) follows from the above estimate setting (µ, d′) = (ν, 0). To prove the second inequality observe that, since |v| > R ≥ 2Λ > 12ε by hypothesis and |v| ≤ |zν,0(k)| by (4.4.3), on the one hand we have 1 4 |v| ≤ 1112 |v| = |v| − 112 |v| ≤ |v| − 16Λ ≤ |v| − ε ≤ |zν,0(k)| − |k1 + i(−1)νk2| ≤ |zν,0(k)− k1 − i(−1)νk2| = 2|k2|. On the other hand, since |zν,0(k)| < 3|v| by (4.4.3), |k2| = |2i(−1)νk2| = |k1 + i(−1)νk2 − (k1 − i(−1)νk2)| = |k1 + i(−1)νk2 − zν,0(k)| ≤ ε+ 3|v| ≤ 4|v|. Combining these estimates we obtain the second inequality of part (i). (ii) Similarly, in view of (4.4.3), if k ∈ Tµ(d′) \ KR for (µ, d′) ∈ {(ν, 0), (ν ′, d)} then 1 |zν,0(k)| ≤ 1 |v| ≤ 3 |zν,0(k)| and 1 |zν′,d(k)| ≤ 1 |v| ≤ 3 |zν′,d(k)| . (4.4.4) 69 These are the first two inequalities of part (ii). Now, since zν′,d(k) = k1 − i(−1)ν′k2 + d1 − i(−1)ν′d2 = zν′,0(k) + d1 − i(−1)ν′d2 = wν,0(k) + d1 − i(−1)ν′d2, |wν,0(k)| < ε, and |d1 − i(−1)ν′d2| = |d|, it follows that |zν′,d(k)| − ε ≤ |d| ≤ |zν′,d(k)|+ ε. Furthermore, by (4.4.4), ε < Λ 6 ≤ |v| 12 ≤ |zν′,d(k)| 12 . Thus, 1 2 |zν′,d(k)| ≤ |d| ≤ 2|zν′,d(k)|. This yields the third inequality of part (ii) and completes the proof. Proof of Lemma 4.4.1 Proof of Lemma 4.4.1. We consider all cases at the same time. Therefore, we have either hypothesis (i) with (µ, d′) = (ν, 0) or hypothesis (ii) with (µ, d′) ∈ {(ν, 0), (ν ′, d)}. Note that either (ν, ν ′) = (1, 2) or (ν, ν ′) = (2, 1). Step 1 Recall the change of variables wµ,d′(k) = k1 + d′1 + i(−1)µ(k2 + d′2), zµ,d′(k) = k1 + d′1 − i(−1)µ(k2 + d′2), and set G′1 := { b ∈ G′ | |b− d′| < 14R } , G′2 := { b ∈ G′ | |b− d′| ≥ 14R } . Observe that G′ = G′1 ∪G′2. By Proposition 4.2.1, G′1, G ′ 2 ⊂ G′ = Γ# \G = {b ∈ Γ# | |Nb(k)| ≥ ε|v|}. 70 Furthermore, by Proposition 4.4.4, for (µ, d′) = (ν, 0) if (i) or (µ, d′) ∈ {(ν, 0), (ν ′, d)} if (ii), 1 |v| ≤ 3 |zµ,d′ | . Thus, using the above remarks and observing the definition of G′2 we have |R1(k)| := ∣∣∣∣∣ ∑ b∈G′1 ∑ c∈G′2 f(d′ − b) Nb(k) (R−1G′G′)b,c g(c− d′) ∣∣∣∣∣ (4.4.5) ≤ 1 ε|v|‖R −1 G′G′‖ ∑ b∈G′1 |f(d′ − b)| ∑ c∈G′2 |c− d′|2 |c− d′|2 |g(c− d ′)| ≤ 1 ε|v|‖R −1 G′G′‖ ‖f‖l1 16 R2 ‖c2g(c)‖l1 ≤ Cε,f,g |zµ,d′ |R2 and |R2(k)| := ∣∣∣∣∣ ∑ b∈G′2 ∑ c∈G′ f(d′ − b) Nb(k) (R−1G′G′)b,c g(c− d′) ∣∣∣∣∣ (4.4.6) ≤ 1 ε|v|‖R −1 G′G′‖ ∑ b∈G′2 |d′ − b|2 |d′ − b|2 |f(d ′ − b)| ∑ c∈G′ |g(c− d′)| ≤ 1 ε|v|‖R −1 G′G′‖ ‖|b|2f(b)‖l1 16 R2 ‖g‖l1 ≤ Cε,f,g |zµ,d′ |R2 . Hence, Φd′,d′(k) = [ ∑ b,c∈G′1 + ∑ b∈G′1 c∈G′2 + ∑ b∈G′2 c∈G′ ] f(d′ − b) Nb(k) (R−1G′G′)b,c g(c− d′) = ∑ b,c∈G′1 f(d′ − b) Nb(k) (R−1G′G′)b,c g(c− d′) +R1(k) +R2(k) (4.4.7) with |R1(k) +R2(k)| ≤ Cε,f,g|zµ,d′ |R2 . (4.4.8) Now, if we set TG′G′ := piG′ −RG′G′ and recall the convergent series expansion R−1G′G′ = (piG′ − TG′G′)−1 = ∞∑ j=0 T jG′G′ , we can write∑ b,c∈G′1 f(d′ − b) Nb(k) (R−1G′G′)b,c g(c− d′) = ∞∑ j=0 ∑ b,c∈G′1 f(d′ − b) Nb(k) (T jG′G′)b,c g(c− d′). (4.4.9) 71 Note, the above equality is fine because G′1 is finite set. Let G′3 := {b ∈ G′ | |b− d′| < 12R}, G′4 := {b ∈ G′ | |b− d′| ≥ 12R}. Again, observe that G′ = G′3 ∪G′4. Thus, we can break TG′G′ into TG′G′ = piG′TpiG′ = (piG′3 + piG′4)T (piG′3 + piG′4) = T33 + T43 + T34 + T44, where Tij := piGiTpiGj for i, j ∈ {3, 4}. Using this decomposition we are able to prove the following. Proposition 4.4.5. Under the hypotheses of Lemma 4.4.1 we have ∞∑ j=0 ∑ b,c∈G′1 f(d′ − b) Nb(k) (T jG′G′)b,c g(c− d′) = ∞∑ j=0 ∑ b,c∈G′1 f(d′ − b) Nb(k) (T j33)b,c g(c− d′) +R3(k) with R3(k) given by (4.4.34) and |R3(k)| ≤ CΛ,f,g|zµ,d′ |R2 . (4.4.10) This proposition will be proved below. Combining this with (4.4.7) and (4.4.9) we obtain Φd′,d′(k) = ∞∑ j=0 ∑ b,c∈G′1 f(d′ − b) Nb(k) (T j33)b,c g(c− d′) + 3∑ j=1 Rj(k). (4.4.11) Step 2 We now look in detail to the operator T33 and its powers T j 33. Recall that θµ(b) = 1 2 ((−1)µb2 + ib1) and set µ′ := µ− (−1)µ so that (−1)µ = −(−1)µ′ . Then, Nb(k) = Nb,µ(k)Nb,µ′(k) = (k1 + b1 + i(−1)µ(k2 + b2))(k1 + b1 + i(−1)µ′(k2 + b2)) = (k1 + d′1 + i(−1)µ(k2 + d′2) + b1 − d′1 + i(−1)µ(b2 − d′2)) × (k1 + d′1 + i(−1)µ ′ (k2 + d′2) + b1 − d′1 + i(−1)µ ′ (b2 − d′2)) = (k1 + d′1 + i(−1)µ(k2 + d′2) + b1 − d′1 − i(−1)µ ′ (b2 − d′2)) × (k1 + d′1 − i(−1)µ(k2 + d′2) + b1 − d′1 − i(−1)µ(b2 − d′2)) = (wµ,d′ − 2iθµ′(b− d′))(zµ,d′ − 2iθµ(b− d′)). 72 Extend the definition of θµ(y) to any y ∈ C2. Thus, 2(k + d′) · Â(b− c) = 2(k1 + d′1)Â1(b− c) + 2(k2 + d′2)Â2(b− c) = (wµ,d′ + zµ,d′)Â1(b− c) + (i(−1)µ(zµ,d′ − wµ,d′))Â2(b− c) = (Â1(b− c)− i(−1)µÂ2(b− c))wµ,d′ + (Â1(b− c) + i(−1)µÂ2(b− c))zµ,d′ = −2iθµ(Â(b− c))wµ,d′ − 2iθµ′(Â(b− c)) zµ,d′ . Hence, Tb,c = 1 Nc(k) (2(c+ k) · Â(b− c)− q̂(b− c)) = 2(c− d′) · Â(b− c)− q̂(b− c) + 2(k + d′) · Â(b− c) (wµ,d′ − 2iθµ′(c− d′))(zµ,d′ − 2iθµ(c− d′)) = Xb,c + Yb,c, (4.4.12) where Xb,c := 2(c− d′) · Â(b− c)− q̂(b− c)− 2iθµ(Â(b− c))wµ,d′ (wµ,d′ − 2iθµ′(c− d′))(zµ,d′ − 2iθµ(c− d′)) , (4.4.13) Yb,c := −2iθµ′(Â(b− c)) zµ,d′ (wµ,d′ − 2iθµ′(c− d′))(zµ,d′ − 2iθµ(c− d′)) . (4.4.14) Let X and Y be the operators whose matrix elements are, respectively, Xb,c and Yb,c. Set X33 := piG′3XpiG′3 and Y33 := piG′3Y piG′3 . We next prove the following estimates, ‖X33‖ ≤ ( 20‖Â‖l1 + 4 Λ ‖q̂‖l1 ) 1 |zµ,d′ |R < 1 3 , ‖Y33‖ ≤ 8Λ‖θµ′(Â)‖l1 < 1 14 , (4.4.15) where |zµ,d′ |R := 2|zµ,d′ | −R. First observe that the “vector” b ∈ Γ# has the same length as the complex number 2iθµ(b): |b| = |(b1, b2)| = |b1 + i(−1)µb2| = |2iθµ(b)|. (4.4.16) Thus, for b ∈ G′3, |2iθµ(b− d′)| R = |b− d′| R < 1 2 . Consequently, 1 |zµ,d′ − 2iθµ(b− d′)| ≤ 1 |zµ,d′ | − |2iθµ(b− d′)| < 1 |zµ,d′ | − 12R = 2 |zµ,d′ |R . (4.4.17) 73 Furthermore, for b ∈ G′, 1 |wµ,d′ − 2iθµ′(b− d′)| ≤ 1 |b− d′| − |wµ,d′ | ≤ 1 |b− d′| − ε (4.4.18) ≤ 1 2Λ− Λ = 1 Λ . (4.4.19) Here we have used that |wµ,d′ | < ε < Λ and |b − d′| ≥ 2Λ for all b ∈ G′. Using again that ε < Λ ≤ |c− d′|/2 for all c ∈ G′ we have |c− d′| |c− d′| − ε < 2. (4.4.20) Finally recall that ε Λ < 1 6 and 1 |zµ,d′ | ≤ 1 |v| < 1 R , (4.4.21) where the last inequality follows from Proposition 4.4.4 since |v| > R by hypothesis. Then, using the above inequalities and Proposition 4.1.2, the bounds (4.4.15) for ‖X33‖ and ‖Y33‖ follow from the estimates sup c∈G′3 ∑ b∈G′3 + sup b∈G′3 ∑ c∈G′3 |Xb,c| ≤ sup c∈G′3 ∑ b∈G′3 + sup b∈G′3 ∑ c∈G′3 2|c− d′| |Â(b− c)|+ |q̂(b− c)|+ |2iθµ(Â(b− c))| |wµ,d′ | |wµ,d′ − 2iθµ′(c− d′)| |zµ,d′ − 2iθµ(c− d′)| (now apply (4.4.16), (4.4.17) and |wµ,d′ | < ε) ≤ 2|zµ,d′ |R sup c∈G′3 ∑ b∈G′3 + sup b∈G′3 ∑ c∈G′3 [ 2|c− d′| |Â(b− c)| |wµ,d′ − 2iθµ′(c− d′)| + |q̂(b− c)|+ ε√2 |Â(b− c)| |wµ,d′ − 2iθµ′(c− d′)| ] (now apply (4.4.18) and (4.4.19)) ≤ 2|zµ,d′ |R sup c∈G′3 ∑ b∈G′3 + sup b∈G′3 ∑ c∈G′3 [2|c− d′| |Â(b− c)| |c− d′| − ε + |q̂(b− c)|+ ε√2 |Â(b− c)| Λ ] (now apply (4.4.20), (4.4.21) and (4.4.2)) ≤ 2|zµ,d′ |R 2 [[ 4 + ε √ 2 Λ ] ‖Â‖l1 + ‖q̂‖l1 Λ ] ≤ [ 20‖Â‖l1 + 4‖q̂‖l1 Λ ] 1 |zµ,d′ |R ≤ [ 20‖Â‖l1 + 4‖q̂‖l1 Λ ] 1 R < 1 7 + 1 4 = 1 3 74 and sup c∈G′3 ∑ b∈G′3 + sup b∈G′3 ∑ c∈G′3 |Yb,c| ≤ sup c∈G′3 ∑ b∈G′3 + sup b∈G′3 ∑ c∈G′3 |2iθµ′(Â(b− c))| |zµ,d′ | |wµ,d′ − 2iθµ′(c− d′)| |zµ,d′ − 2iθµ(c− d′)| (now apply (4.4.16), (4.4.17) and (4.4.19)) ≤ 2 Λ sup c∈G′3 ∑ b∈G′3 + sup b∈G′3 ∑ c∈G′3 [ |2θµ′(Â(b− c))| |zµ,d′ | |zµ,d′ |R ] ≤ 8 Λ ‖θµ′(Â)‖l1 ≤ 8 √ 2 Λ ‖Â‖l1 ≤ 16 √ 2 63 ε Λ < 16 √ 2 63 1 6 < 1 14 . Step 3 We now look in detail to T j33. For each integer j ≥ 1 write T j33 = (X33 + Y33) j = Zj +Wj + Y j 33, (4.4.22) where Wj is the sum of the j terms containing only one factor X33 and j − 1 factors Y33, Wj := X33Y33 · · ·Y33 + Y33X33Y33 · · ·Y33 + · · ·+ Y33 · · ·Y33X33 = j∑ m=1 (Y33)m−1X33(Y33)j−m, and Zj := (X33 + Y33)j −Wj − Y j33. In view of (4.4.15) we have ‖Y33‖j ≤ ( 1 14 )j , ‖Wj‖ ≤ j‖X33‖ ‖Y33‖j−1 ≤ CΛ,A,q|zµ,d′ |R j ( 1 14 )j−1 , ‖Zj‖ ≤ (2j − j − 1) ‖X33‖2 ( 1 3 )j−2 ≤ CΛ,A,q|zµ,d′ |2R ( 2 3 )j . Hence, the series S := ∞∑ j=0 Y j33 = (I − Y33)−1, W := ∞∑ j=1 Wj and Z := ∞∑ j=2 Zj (4.4.23) 75 converge, and the operator norm of W and Z decay with respect to |zµ,d′ |. Indeed, ‖S‖ ≤ ∞∑ j=0 ‖Y33‖j ≤ ∞∑ j=0 ( 1 14 )j < C, ‖W‖ ≤ ∞∑ j=1 ‖Wj‖ ≤ C ′Λ,A,q 2|zµ,d′ | −R ∞∑ j=1 j ( 1 14 )j−1 < CΛ,A,q |zµ,d′ |R , ‖Z‖ ≤ ∞∑ j=2 ‖Zj‖ ≤ C ′Λ,A,q |zµ,d′ |2R ∞∑ j=2 ( 2 3 )j ≤ CΛ,A,q|zµ,d′ |2R . Thus, we have the expansion ∞∑ j=0 T j33 = S +W + Z. Step 4 Consequently, ∞∑ j=0 ∑ b,c∈G′1 f(d′ − b) Nb(k) (T j33)b,c g(c− d′) = ∑ b,c∈G′1 f(d′ − b) (S +W + Z)b,c g(c− d′) (wµ,d′ − 2iθµ′(b− d′))(zµ,d′ − 2iθµ(b− d′)) = α(1)µ,d′ + α (2) µ,d′ +R4, (4.4.24) where α (1) µ,d′(k) := ∑ b,c∈G′1 f(d′ − b)Sb,c(k) g(c− d′) (wµ,d′(k)− 2iθµ′(b− d′))(zµ,d′(k)− 2iθµ(b− d′)) , α (2) µ,d′(k) := ∑ b,c∈G′1 f(d′ − b)Wb,c(k) g(c− d′) (wµ,d′(k)− 2iθµ′(b− d′))(zµ,d′(k)− 2iθµ(b− d′)) (4.4.25) and R4(k) := ∑ b,c∈G′1 f(d′ − b)Zb,c(k) g(c− d′) (wµ,d′(k)− 2iθµ′(b− d′))(zµ,d′(k)− 2iθµ(b− d′)) . (4.4.26) By a short calculation as in (4.4.33), using (4.4.17) and (4.4.19) we find that |α(1)µ,d′(k)| ≤ 1 Λ 2 2|zµ,d′ | −R‖f‖l1‖g‖l1‖S‖ ≤ CΛ,f,g |zµ,d′ |R , |α(2)µ,d′(k)| ≤ 1 Λ 2 2|zµ,d′ | −R‖f‖l1‖g‖l1‖W‖ ≤ CΛ,A,q,f,g |zµ,d′ |2R , |R4(k)| ≤ 1Λ 2 2|zµ,d′ | −R‖f‖l1‖g‖l1‖Z‖ ≤ CΛ,A,q,f,g |zµ,d′ |3R . (4.4.27) Hence, recalling (4.4.11) we conclude that Φd′,d′ = α (1) µ,d′ + α (2) µ,d′ + α (3) µ,d′ , 76 where α (3) µ,d′(k) := 4∑ j=1 Rj(k). (4.4.28) Furthermore, in view of (4.4.8), (4.4.10) and (4.4.27), since 1 |zµ,d′ |3R = 1 (2|zµ,d′ | −R)3 < 1 |zµ,d′ |R2 , for 1 ≤ j ≤ 2 we have |α(j)µ,d′(k)| ≤ Cj |zµ,d′(k)|jR and |α(3)µ,d′(k)| ≤ C3 |zµ,d′(k)|R2 , where Cj = Cj;Λ,A,q,f,g and C3 = C3;ε,Λ,A,q,f,g are constants. This proves the main statement of the lemma. Finally observe that, since G′3 is a finite set, the matrices X33 and Y33 are analytic in k because their matrix elements are analytic functions of k. (Note, the functions wµ,d′(k) and zµ,d′(k) are analytic.) Consequently, the matrices Wj and Zj are also analytic and so are Sb,c, Wb,c and Zb,c because the series (4.4.23) converge uniformly with respect to k. Thus, all the functions α(j)µ,d′(k) are analytic in the region under consideration. This completes the proof of the lemma. We now prove Proposition 4.4.5, which was used in the above proof. Proof of Proposition 4.4.5. Step 1 Recall that TG′G′ = T33 + T34 + T43 + T44 with Tij = piG′iTpiG′j , and set X (0) 33 := 0, Y (0) 34 := T34, W (0) 43 := T43 and Z (0) 44 := T44. First observe that T 2G′G′ = T 2 33 +X (1) 33 + Y (1) 34 +W (1) 43 + Z (1) 44 , where X (1) 33 := T33X (0) 33 + T34W (0) 43 : L 2 G′3 → L2G′3 , Y (1) 34 := T33Y (0) 34 + T34Z (0) 44 : L 2 G′3 → L2G′4 , W (1) 43 := T43T33 + T43X (0) 33 + T44W (0) 43 : L 2 G′4 → L2G′3 , Z (1) 44 := T43Y (0) 34 + T44Z (0) 44 : L 2 G′4 → L2G′4 . 77 Now suppose that T jG′G′ = T j 33 +X (j−1) 33 + Y (j−1) 34 +W (j−1) 43 + Z (j−1) 44 with X (j−1) 33 : L 2 G′3 → L2G′3 , Y (j−1) 34 : L 2 G′3 → L2G′4 , W (j−1) 43 : L 2 G′4 → L2G′3 , Z (j−1) 44 : L 2 G′4 → L2G′4 . It is straightforward to verify that T j+1G′G′ = T j+1 33 +X (j) 33 + Y (j) 34 +W (j) 43 + Z (j) 44 , (4.4.29) where X (j) 33 := T33X (j−1) 33 + T34W (j−1) 43 : L 2 G′3 → L2G′3 , Y (j) 34 := T33Y (j−1) 34 + T34Z (j−1) 44 : L 2 G′3 → L2G′4 , W (j) 43 := T43T j 33 + T43X (j−1) 33 + T44W (j−1) 43 : L 2 G′4 → L2G′3 , Z (j) 44 := T43Y (j−1) 34 + T44Z (j−1) 44 : L 2 G′4 → L2G′4 . (4.4.30) Thus, it follows by induction that (4.4.29) and (4.4.30) hold for any j ≥ 0. Step 2 Since piG′1piG′4 = piG′4piG′1 = 0 and piG′1piG′3 = piG′3piG′1 = piG′1 , substituting (4.4.29) into the sum below for the terms where j ≥ 1 we have, recalling that X(0)33 = 0, ∞∑ j=0 ∑ b,c∈G′1 f(d′ − b) Nb(k) (T jG′G′)b,c g(c− d′) = 0∑ j=0 + ∞∑ j=1 (· · · ) = ∞∑ j=0 ∑ b,c∈G′1 f(d′ − b) Nb(k) (T j33)b,c g(c− d′) + ∞∑ j=1 ∑ b,c∈G′1 f(d′ − b) Nb(k) (X(j−1)33 )b,c g(c− d′)︸ ︷︷ ︸ ∞∑ j=1 ∑ b,c∈G′1 f(d′ − b) Nb(k) (X(j)33 )b,c g(c− d′) . (4.4.31) Now recall from (4.4.17) and (4.4.19) that, for all b ∈ G′3, 1 |Nb(k)| ≤ 2 Λ 1 |zµ,d′ |R , (4.4.32) 78 and observe that G′1 ⊂ G′3. Let M be either TG′G′ or T33. Then, the estimate∣∣∣∣∣ ∑ b,c∈G′1 f(d′ − b) Nb(k) (Mj)b,c g(c− d′) ∣∣∣∣∣ = ∣∣∣∣∣ ∑ b∈G′1 f(d′ − b) Nb(k) ∑ c∈G′1 〈 eib·x |Γ|1/2 ,M j e ic·x |Γ|1/2 〉 g(c− d′) ∣∣∣∣∣ ≤ 1|Γ| ∑ b∈G′1 |f(d′ − b)| |Nb(k)| ∑ c∈G′1 ‖eib·x‖L2‖Mj‖ ‖eic·x‖L2 |g(c− d′)| ≤ 2 Λ 1 |zµ,d′ |R ‖f‖l1‖g‖l1‖M‖ j (4.4.33) implies that the left hand side and the first term on the right hand side of (4.4.31) converge because ‖M‖ < 17/18. Thus, the last term in (4.4.31) also converges. Hence, we are left to show that R3(k) := ∞∑ j=1 ∑ b,c∈G′1 f(d′ − b) Nb(k) (X(j)33 )b,c g(c− d′) (4.4.34) obeys |R3(k)| ≤ CΛ,f,g|zµ,d′ |R2 . In order to do this we need the following inequality, which we prove later. Proposition 4.4.6. Consider a constant β ≥ 0 and suppose that ‖(1 + |b|β)q̂(b)‖l1 < ∞ and ‖(1 + |b|β)Â(b)‖l1 < 2ε/63. Suppose further that |v| > 2ε‖(1 + |b|β)Â(b)‖l1. Then, for any B,C ⊂ G′ and m ≥ 1, ‖piBTmG′G′piC‖ ≤ (1 + (2Λ)β−dβedβemdβe−1) ( 17 18 )m sup b∈B c∈C 1 1 + |b− c|β , where dβe is the smallest integer greater or equal than β. Step 3 Now observe that, if b ∈ G′1 and c ∈ G′4 then |b− c| = |b− d′ − (c− d′)| ≥ |c− d′| − |b− d′| ≥ R 2 − R 4 = R 4 . Thus, applying the last proposition with β = 2 and recalling that G′3 ⊂ G′, for m ≥ 0 we have ‖piG′1Tm33T34‖ ≤ ‖piG′1TmG′G′TG′G′4‖ = ‖piG′1Tm+1G′G′ piG′4‖ ≤ 3(m+ 1) 1 + 116R 2 ( 17 18 )m+1 . Furthermore, since piG′4piG′3 = piG′4piG′1 = 0 and piG′3piG′1 = piG′1 , from (4.4.29) we obtain W (j) 43 piG′1 = piG′4T j+1 G′G′piG′3piG′1 = piG′4T j+1 G′G′piG′1 . 79 Hence, ‖W (j)43 piG′1‖ = ‖piG′4T j+1 G′G′piG′1‖ ≤ ‖TG′G′‖j+1 < ( 17 18 )j+1 . Therefore, for 0 ≤ m < j, ‖piG′1Tm33T34W (j−m−1)piG′1‖ ≤ ‖piG′1Tm33T34‖ ‖W (j−m−1) 43 piG′1‖ ≤ 3(m+ 1) 1 + 116R 2 ( 17 18 )j+1 . Iterating the first expression in (4.4.30) we find that X (j) 33 = T34W (j−1) 43 + T33X (j−1) 33 = T34W (j−1) 43 + T33T34W (j−2) 43 + T 2 33X (j−2) 33 ... = T34W (j−1) 43 + T33T34W (j−2) 43 + · · ·+ T j−233 T34W (1)43 + T j−133 T34W (0)43 = j−1∑ m=0 Tm33T34W (j−m−1) 43 . (4.4.35) Thus, using the above inequality, ‖piG′1X (j) 33 piG′1‖ = ∥∥∥∥∥ j−1∑ m=0 piG′1T m 33T34W (j−m−1) 43 piG′1 ∥∥∥∥∥ ≤ j−1∑ m=0 ‖piG′1Tm33T34W (j−m−1) 43 piG′1‖ ≤ 3 1 + 116R 2 ( 17 18 )j+1 j−1∑ m=0 (m+ 1) = 3 2 + 18R 2 (j2 + j) ( 17 18 )j+1 . Consequently, ∥∥∥∥∥∥piG′1 ∞∑ j=1 X (j) 33 piG′1 ∥∥∥∥∥∥ ≤ ∞∑ j=1 ‖piG′1X (j) 33 piG′1‖ ≤ 3 2 + 18R 2 ∞∑ j=1 (j2 + j) ( 17 18 )j+1 ≤ C R2 , where C is an universal constant. Finally, using this and (4.4.32), since |zµ,d′ | ≤ 3|v| we have |R3(k)| = ∣∣∣∣∣∣ ∑ b,c∈G′1 f(d′ − b) Nb(k) ∞∑ j=1 X (j) 33 b,c g(c− d′) ∣∣∣∣∣∣ ≤ 2 Λ|v|‖f‖l1 ∥∥∥∥∥∥piG′1 ∞∑ j=1 X (j) 33 piG′1 ∥∥∥∥∥∥ ‖g‖l1 ≤ 6CΛ ‖f‖l1‖g‖l1 1|zµ,d′ |R2 . In view of (4.4.31) and (4.4.34) this completes the proof. 80 We now prove Proposition 4.4.6, which was used above and left behind without proof. This is the last step we need to finish the proof of Lemma 4.4.1 indeed. Proof of Proposition 4.4.6. For any b, c ∈ Γ# set Qb,c := (1 + |b− c|β)Tb,c. We first claim that, for any B,C ⊂ G′, sup b∈B ∑ c∈C |Qb,c| < 1718 and supc∈C ∑ b∈B |Qb,c| < 1718 . (4.4.36) In fact, using the bounds (4.1.4), (4.1.5) and (4.2.2), namely, for all b ∈ G′, 1 |Nb(k)| ≤ 1 ε|v| , |b| |Nb(k)| ≤ 4 ε and |k| ≤ |u|+ |v| ≤ 3|v|, it follows that sup b∈B ∑ c∈C |Qb,c| = sup b∈B ∑ c∈C (1 + |b− c|β) ∣∣∣∣∣ q̂(b− c)Nc(k) − 2c · Â(b− c)Nc(k) − 2k · Â(b− c)Nc(k) ∣∣∣∣∣ ≤ ‖(1 + |b|β)q̂(b)‖l1 1 ε|v| + 14 ε ‖(1 + |b|β)Â(b)‖l1 < 1 2 + 4 9 = 17 18 and sup c∈C ∑ b∈B |Qb,c| = sup c∈C ∑ b∈B (1 + |b− c|β) ∣∣∣∣∣ q̂(b− c)Nc(k) − 2c · Â(b− c)Nc(k) − 2k · Â(b− c)Nc(k) ∣∣∣∣∣ ≤ ‖(1 + |b|β)q̂(b)‖l1 1 ε|v| + 14 ε ‖(1 + |b|β)Â(b)‖l1 < 1 2 + 4 9 = 17 18 . Furthermore, since |Tb,c| ≤ |Qb,c| for all b, c ∈ Γ#, for any integer m ≥ 1 we have sup b∈B ∑ c∈C |(TmBC)b,c| < ( 17 18 )m and sup c∈C ∑ b∈B |(TmBC)b,c| < ( 17 18 )m . Now, let p be the smallest integer greater or equal than β, and for any integer m ≥ 1 and any ξ0, ξ1, . . . , ξm ∈ Γ#, let b = ξ0 and c = ξm. Then, |b− c|β = (2Λ)β [ |b− c| 2Λ ]β ≤ (2Λ)β [ |b− c| 2Λ ]p = (2Λ)β (2Λ)p m∑ i1,...,ip=1 |ξi1−1 − ξi1 | · · · |ξip−1 − ξip | ≤ (2Λ)β−p m∑ i1,...,ip=1 max {|ξi1−1 − ξi|p, . . . , |ξip−1 − ξip |p} ≤ (2Λ)β−p m∑ i1,...,ip=1 (|ξi1−1 − ξi1 |p + · · ·+ |ξip−1 − ξip |p) = (2Λ)β−ppmp−1 m∑ i=1 |ξi−1 − ξi|p ≤ (2Λ)β−ppmp−1 m∏ i=1 (1 + |ξi−1 − ξi|p). (4.4.37) 81 To simplify the notation write s := sup b∈B c∈C 1 1 + |b− c|β . Hence, sup b∈B ∑ c∈C |(TmG′G′)b,c| ≤ sup b∈B c∈C 1 1 + |b− c|β supb∈B ∑ c∈C (1 + |b− c|β)|(TmG′G′)b,c| ≤ s sup b∈B ∑ c∈C |(TmG′G′)b,c|+ (2Λ)β−ppmp−1 sup b∈B ∑ ξ1∈G′ (1 + |b− ξ1|β)|Tb,ξ1 | × ∑ ξ2∈G′ (1 + |ξ1 − ξ2|2)|Tξ1,ξ2 | · · · ∑ c∈C (1 + |ξm−1 − c|2)|Tξm−1,c| ≤ s (17 18 )m + (2Λ)β−ppmp−1 sup b∈B ∑ ξ1∈G′ (1 + |b− ξ1|2)|Tb,ξ1 | × sup ξ1∈G′ ∑ ξ2∈G′ (1 + |ξ1 − ξ2|2)|Tξ1,ξ2 | · · · sup ξm−1∈G′ ∑ c∈C (1 + |ξm−1 − c|2)|Tξm−1,c| = s (17 18 )m + (2Λ)β−ppmp−1 sup b∈B ∑ ξ1∈G′ |Qb,ξ1 | · · · sup ξm−1∈G′ ∑ c∈C |Qξm−1,c| ≤ s (1 + (2Λ)β−ppmp−1) ( 17 18 )m . Similarly, sup c∈C ∑ b∈B |(TmG′G′)b,c| ≤ sup b∈B c∈C 1 1 + |b− c|2 supc∈C ∑ b∈B (1 + |b− c|2)|(TmG′G′)b,c| ≤ s sup c∈C ∑ b∈B |(TmG′G′)b,c|+ (2Λ)β−ppmp−1 sup c∈C ∑ ξm−1∈G′ (1 + |ξm−1 − c|2)|Tξm−1,c | × ∑ ξm−2∈G′ (1 + |ξm−2 − ξm−1|2)|Tξm−2,ξm−1 | · · · ∑ b∈B (1 + |b− ξ1|2)|Tb,ξ1 | ≤ s (17 18 )m + (2Λ)β−ppmp−1 sup c∈C ∑ ξm−1∈G′ (1 + |ξm−1 − c|2)|Tξm−1,c | × sup ξm−1∈G′ ∑ ξm−2∈G′ (1 + |ξm−2 − ξm−1|2)|Tξm−2,ξm−1 | · · · sup ξ1∈G′ ∑ b∈B (1 + |b− ξ1|2)|Tb,ξ1 | ≤ s (1 + (2Λ)β−ppmp−1) ( 17 18 )m . 82 Therefore, by Proposition 4.1.2, ‖piBTmG′G′piC‖ ≤ (1 + (2Λ)β−dβedβemdβe−1) ( 17 18 )m sup b∈B c∈C 1 1 + |b− c|β , where dβe is the smallest integer greater or equal than β. This is the desired inequality. Proof of Lemma 4.4.2 Proof of Lemma 4.4.2. To simplify the notation write w = wµ,d′ , z = zµ,d′ and |z|R = 2|z| −R. First observe that 1 w − 2iθµ′(c− d′) = −1 2iθµ′(c− d′) + w 2iθµ′(c− d′)(w − 2iθµ′(c− d′)) , so that z Nc(k) = z (w − 2iθµ′(c− d′))(z − 2iθµ(c− d′)) = 1 w − 2iθµ′(c− d′) ( 1 + 2iθµ(c− d′) z − 2iθµ(c− d′) ) = 1 w − 2iθµ′(c− d′) + 2iθµ(c− d′) w − 2iθµ′(c− d′) 1 z − 2iθµ(c− d′) = −1 2iθµ′(c− d′) + w 2iθµ′(c− d′)(w − 2iθµ′(c− d′)) + 2iθµ(c− d′) w − 2iθµ′(c− d′) 1 z − 2iθµ(c− d′) =: η(0)c + η (w) c + η (z) c , where, in view of (4.4.17) to (4.4.20), since |w| < ε, |η(0)c | ≤ 1 2Λ , |η(w)c | ≤ ε 2Λ2 and |η(z)c | ≤ 4 |z|R . Hence, Yb,c = −2iθµ′(Â(b− c))z Nc(k) = −2iθµ′(Â(b− c))η(0)c − 2iθµ′(Â(b− c))η(w)c − 2iθµ′(Â(b− c))η(z)c =: Y (0)b,c + Y (w) b,c + Y (z) b,c . Let Y ( · ) be the operator whose matrix elements are Y ( · )b,c and set Y ( · ) 33 := piG′3Y ( · )piG′3 . Then, similarly as we estimated ‖Y33‖, using (4.4.17) to (4.4.20) and Proposition 4.1.2, it 83 follows easily that ‖Y (0)33 ‖ ≤ 1 2Λ ‖θµ′(Â)‖l1 , ‖Y (w)33 ‖ ≤ ε 2Λ2 ‖θµ′(Â)‖l1 , ‖Y (z)33 ‖ ≤ 4 |z|R ‖θµ ′(Â)‖l1 . Furthermore, S = (I − Y33)−1 = 1 + (1− Y33)−1Y33 = 1 + SY33 = 1 + (1 + SY33)Y33 = 1 + Y33 + SY 233 = 1 + Y (0)33 + Y (w) 33 + Y (z) 33 + SY 2 33, where, recalling (4.4.15), ‖SY 233‖ ≤ ‖(1− Y33)−1‖ ‖Y33‖2 ≤ ‖Y33‖2 1− ‖Y33‖ < 14 13 ( 8 Λ )2 ‖θµ′(Â)‖2l1 . Combining all this we have z Sb,c Nb(k) = (η(0)b + η (w) b + η (z) b )Sb,c = (η(0)b + η (w) b )(δb,c + Y (0) b,c + Y (w) b,c + Y (z) b,c + (SY 2 33)b,c) + η (z) b Sb,c = [ η (0) b (δb,c + Y (0) b,c ) ] + [ η (0) b Y (w) b,c + η (w) b (δb,c + Y (0) b,c + Y (w) b,c ) ] + [ (η(0)b + η (w) b )(SY 2 33)b,c ] + [ (η(0)b + η (w) b )Y (z) b,c + η (z) b Sb,c ] =: K(0)b,c +K (1) b,c +K (2) b,c +K (3) b,c with |K(0)b,c | ≤ 1 2Λ ( 1 + 1 2Λ ‖θµ′(Â)‖l1 ) , |K(1)b,c | ≤ ε 4Λ3 ‖θµ′(Â)‖l1 + ε 2Λ2 ( 1 + 1 2Λ ‖θµ′(Â)‖l1 + ε Λ2 ‖θµ′(Â)‖l1 ) < ε 2Λ2 ( 1 + 7 6Λ ‖θµ′(Â)‖l1 ) , |K(2)b,c | ≤ 1 Λ ( 8 Λ )2 ‖θµ′(Â)‖2l1 < 64 Λ3 ‖θµ′(Â)‖2l1 , |K(3)b,c | ≤ 3 2Λ ‖θµ′(Â)‖l1 4 |z|R + 14 13 4 |z|R < CΛ,A |z|R for all b, c ∈ G′3. Here, to estimate |K(1)b,c | we have used that ε < Λ/6. 84 Finally, recalling (4.4.25) and using the above estimates we find that zµ,d′(k)α (1) µ,d′(k) = ∑ b,c∈G′1 f(d′ − b) z Sb,c Nb(k) g(c− d′) = ∑ b,c∈G′1 f(d′ − b) [ 3∑ j=0 K (j) b,c ] g(c− d′) =: α(1,0)µ,d′ + α (1,1) µ,d′ (w(k)) + α (1,2) µ,d′ (k) + α (1,3) µ,d′ (k), (4.4.38) where, in particular, α (1,0) µ,d′ = − ∑ b,c∈G′1 f(d′ − b) 2iθµ′(b− d′) [ δb,c + θµ′(Â(b− c)) θµ′(c− d′) ] g(c− d′). (4.4.39) Furthermore, it follows easily from (4.4.38) that, for 0 ≤ j ≤ 2, |α(1,j)µ,d′ | ≤ Cj with C0 := 1 2Λ ( 1 + 1 2Λ ‖θµ′(Â)‖l1 ) ‖f‖l1‖g‖l1 , C1 := ε 2Λ2 ( 1 + 7 6Λ ‖θµ′(Â)‖l1 ) ‖f‖l1‖g‖l1 , C2 := 64 Λ3 ‖θµ′(Â)‖2l1‖f‖l1‖g‖l1 , (4.4.40) while for j = 3, |α(1,3)µ,d′ | ≤ CΛ,A,f,g 1 |z|R . This completes the proof of the lemma. Proof of Lemma 4.4.3 We first derive the following inequality. Proposition 4.4.7. Let α and δ be constants with 1 < α ≤ 2 and 1 < δ ≤ 2. Suppose that f is a function on Γ# obeying ‖|b|αf(b)‖l1 <∞. Then, for any ξ1, ξ2 ∈ Γ# with ξ1 6= ξ2, ∑ b∈Γ#\{ξ1,ξ2} |f(b− ξ1)| |b− ξ2|δ ≤ C |ξ1 − ξ2|α+δ−2 × 1 if α, δ < 2, ln |ξ1 − ξ2| if α = 2 or δ = 2, where C = CΓ#,α,δ,f is a constant. 85 Proof. We can obtain the desired estimate as follows:∑ b∈Γ#\{ξ1,ξ2} |f(b− ξ1)| |b− ξ2|δ ≤ supb∈Γ# (|b− ξ1|α |f(b− ξ1)|) ∑ b∈Γ#\{ξ1,ξ2} 1 |b− ξ2|δ |b− ξ1|α ≤ ‖|b|αf(b)‖l1 ∑ b∈Γ#\{ξ1,ξ2} 1 |b− ξ2|δ |b− ξ1|α = ‖|b|αf(b)‖l1 ∑ b∈Γ#\{0,ξ1−ξ2} 1 |b|δ |b− (ξ1 − ξ2)|α ≤ ‖|b|αf(b)‖l1 CΓ# ∫ |x|≥Λ |x−(ξ1−ξ2)|≥Λ d2x |x|δ |x− (ξ1 − ξ2)|α (by a rotation in R2 such that ξ1 − ξ2 → |ξ1 − ξ2|(1, 0)) = ‖|b|αf(b)‖l1 CΓ# ∫ |y|≥Λ |y−|ξ1−ξ2|(1,0)|≥Λ d2y |y|δ |y − |ξ1 − ξ2|(1, 0)|α = ‖|b|αf(b)‖l1 CΓ# |ξ1 − ξ2|δ+α−2 ∫ |z|≥Λ/|ξ1−ξ2| |z−(1,0)|≥Λ/|ξ1−ξ2| d2z |z|δ |z − (1, 0)|α ≤ CΓ#,α,δ,f|ξ1 − ξ2|δ+α−2 × 1 if α, δ < 2, ln |ξ1 − ξ2| if α = 2 or δ = 2. We now apply this proposition for proving Lemma 4.4.3. Proof of Lemma 4.4.3. First observe that ‖pi{b}TmG′G′pi{c}‖ = sup ϕ∈L2 ‖ϕ‖L2=1 ‖pi{b}TmG′G′pi{c}ϕ‖L2 = ∥∥∥∥pi{b}TmG′G′ eic·x|Γ|1/2 ∥∥∥∥ L2 = ∥∥∥∥〈 eib·x|Γ|1/2 , TmG′G′ eic·x|Γ|1/2 〉 L2 eib·x |Γ|1/2 ∥∥∥∥ L2 = ∣∣∣∣〈 eib·x|Γ|1/2 , TmG′G′ eic·x|Γ|1/2 〉 L2 ∣∣∣∣ = |(TmG′G′)b,c|. Hence, by Proposition 4.4.6 with β = 2, for all b, c ∈ G′ and m ≥ 1, |(TmG′G′)b,c| = ‖pi{b}TmG′G′pi{c}‖ ≤ (1 + 2m) ( 17 18 )m 1 1 + |b− c|2 . This inequality is also valid for m = 0 because |(T 0G′G′)b,c| = |δb,c| = 1 if b = c 0 if b 6= c ≤ 11 + |b− c|2 . 86 Thus, |Φd′,d′′(k)| = ∣∣∣∣∣∣ ∞∑ m=0 ∑ b,c∈G′ f(d′ − b) Nb(k) (TmG′G′)b,c g(c− d′′) ∣∣∣∣∣∣ ≤ 1 ε|v| [ ∞∑ m=0 (1 + 2m) ( 17 18 )m]∑ b∈G′ |f(d′ − b)| ∑ c∈G′ |g(c− d′′)| 1 + |b− c|2 ≤ C ε|v| ∑ b∈G′ |f(d′ − b)| |g(b− d′′)|+ ∑ c∈G′\{b} |g(c− d′′)| |b− c|2 , (4.4.41) where C is an universal constant. Now, by the triangle inequality, Hölder’s inequality, and since ‖ · ‖l2 ≤ ‖ · ‖l1 ,∑ b∈G′ |f(d′ − b)| |g(b− d′′)| = ∑ b∈G′ |d′ − d′′|2 |d′ − d′′|2 |f(d ′ − b)| |g(b− d′′)| ≤ 4|d′ − d′′|2 ∑ b∈G′ (|d′ − b|2 + |b− d′′|2) |f(d′ − b)| |g(b− d′′)| ≤ 4|d′ − d′′|2 (‖b 2f(b)‖l2‖g‖l2 + ‖f‖l2‖b2g(b)‖l2) ≤ 4|d′ − d′′|2 (‖b 2f(b)‖l1‖g‖l1 + ‖f‖l1‖b2g(b)‖l1) ≤ Cf,g |d′ − d′′|2 . (4.4.42) Furthermore, by Proposition 4.4.7 with α = δ = 2, for any 0 < 1 < 2,∑ c∈G′\{b} |g(c− d′′)| |b− c|2 ≤ CΓ#,g ln |b− d′′| |b− d′′|2 ≤ CΓ#,g,1 |b− d′′|2−1 . Applying this inequality and (4.4.42) to (4.4.41) we obtain |Φd′,d′′(k)| ≤ C ε|v| [ Cf,g |d′ − d′′|2 + CΓ#,g,1 ∑ b∈G′ |f(d′ − b)| |b− d′′|2−1 ] . Applying again Proposition 4.4.7 with α = 2 and δ = 2 − 1 we conclude that, for any 0 < 2 < 2− 1, |Φd′,d′′(k)| ≤ C ε|v| [ Cf,g |d′ − d′′|2 + CΓ#,f,g,1 ln |d′ − d′′| |d′ − d′′|2−1 ] ≤ Cε,Γ#,f,g,1,2|v| |d′ − d′′|2−1−2 . Finally, recall from Proposition 4.4.4(ii) that |zν′,d| < 3|d| and |zν′,d| < 3|v|, observe that |d′ − d′′| = |d|, and set = 1 + 2. Then, for any 0 < < 2, |Φd′,d′′(k)| ≤ Cε,Γ#,f,g,1,2 |d| |d|2−1−2 ≤ Cε,Γ#,f,g, |zν′,d|3− . Choosing = 10−1 we obtain the desired inequality. 87 4.5 Bounds on the derivatives In the last section we expressed Φd′,d′′(k) as a sum of certain functions α (j) µ,d′(k) for k in the ε-tubes with large |v|. In this section we provide bounds for the derivatives of all these functions. We first give all the statements and then the proofs. Our first lemma concerns the derivatives of Φd′,d′′(k). Lemma 4.5.1 (Derivatives of Φd′,d′′(k)). Under Hypothesis 4.4.1, let f and g be functions in l1(Γ#) and suppose either (i) or (ii) where: (i) G = {0} and k ∈ (T0 \ ∪b∈G′Tb) \ KR; (ii) G = {0, d} and k ∈ (T0 ∩ Td) \ KR. Then, for any integers n and m with n+m ≥ 1 and for any d′, d′′ ∈ G,∣∣∣∣ ∂n+m∂kn1 ∂km2 Φd′,d′′(k) ∣∣∣∣ ≤ C|v| , where C is a constant with C = Cε,Λ,A,f,g,m,n if (i) or C = CΛ,A,f,g,m,n if (ii). We now improve the estimate of Lemma 4.5.1(ii) for d′ 6= d′′. Lemma 4.5.2 (Derivatives of Φd′,d′′(k) for d′ 6= d′′). Consider a constant β ≥ 2 and suppose that ‖|b|β q̂(b)‖l1 < ∞ and ‖(1 + |b|β)Â(b)‖l1 < 2ε/63. Let ν ∈ {1, 2} and let f and g be functions on Γ# obeying ‖|b|βf(b)‖l1 < ∞ and ‖|b|βg(b)‖l1 < ∞. Suppose further that G = {0, d} and k ∈ T0 ∩ Td with |v| > 2ε‖|b|β q̂(b)‖l1. Then, for any integers n and m with n+m ≥ 0 and for any d′, d′′ ∈ G with d′ 6= d′′,∣∣∣∣ ∂n+m∂kn1 ∂km2 Φd′,d′′(k) ∣∣∣∣ ≤ C|d|1+β , where C = Cε,Λ,A,f,g,m,n is a constant. Observe that, in particular, this Lemma with m = n = 0 generalizes Lemma 4.4.3. We next have bounds for the derivatives of α(j)µ,d′(k). Lemma 4.5.3 (Derivatives of α(j)µ,d′(k)). Under Hypothesis 4.4.1, let ν ∈ {1, 2} and let f and g be functions in l1(Γ#). Suppose either (i) or (ii) where: (i) G = {0} and k ∈ (Tν(0) \ ∪b∈G′Tb) \ KR; (ii) G = {0, d} and k ∈ (Tν(0) ∩ Tν′(d)) \ KR. 88 Then, there is a constant ρ = ρε,A,q,m,n with ρ ≥ R such that, for |v| ≥ ρ and for (µ, d′) = (ν, 0) if (i) or (µ, d′) ∈ {(ν, 0), (ν ′, d)} if (ii), for any integers n and m with n+m ≥ 1 and for 1 ≤ j ≤ 2,∣∣∣∣ ∂n+m∂kn1 ∂km2 α(j)µ,d′(k) ∣∣∣∣ ≤ Cj(2|zµ,d′(k)| −R)j and ∣∣∣∣ ∂n+m∂kn1 ∂km2 α(3)µ,d′(k) ∣∣∣∣ ≤ C3|zµ,d′(k)|R2 , where Cl = Cl;f,g,Λ,A,q,n,m for 1 ≤ l ≤ 3 are constants. Furthermore, C1;f,g,Λ,A,1,0, C1;f,g,Λ,A,0,1 ≤ 13Λ−2‖f‖l1‖g‖l1 and C1;f,g,Λ,A,1,1 ≤ 65Λ−3‖f‖l1‖g‖l1 . Proof of Lemma 4.5.1 Proof of Lemma 4.5.1. Step 0 When there is no risk of confusion we shall use the same notation to denote an operator or its matrix. Define FBC := [f(b− c)]b∈B,c∈C , GBC := [g(b− c)]b∈B,c∈C , ΦG(k) := [ Φd′,d′′(k;G) ] d′,d′′∈G . Here FBC and GBC are |B| × |C| matrices and ΦG(k) is a |G| × |G| matrix. First observe that ΦG(k) = ∑ b,c∈G′ f(d′ − b) Nb(k) (R−1G′G′)b,c g(c− d′′) d′,d′′∈G can be written as the product of matrices FGG′∆−1k R−1G′G′GG′G. Furthermore, since on L2G′ we have ∆ −1 k R −1 G′G′ = (RG′G′∆k) −1 = H−1k , we can write ΦG(k) as FGG′H−1k GG′G. Hence, ∂n+m ∂kn1 ∂k m 2 ΦG(k) = FGG′ ∂n+mH−1k ∂kn1 ∂k m 2 GG′G. (4.5.1) This is the quantity we want to estimate. Step 1 Let T = T (k) be an invertible matrix. Then applying ∂ m0 ∂k m0 i to the identity TT−1 = I and using the Leibniz rule for ∂ m0 ∂k m0 i (TT−1) we find that ∂m0T−1 ∂km0i = −T−1 m0−1∑ m1=0 ( m0 m1 ) ∂m0−m1T ∂km0−m1i ∂m1T−1 ∂km1i . 89 Iterating this formula m0 − 1 times we obtain ∂m0T−1 ∂km0i = m0∏ j=1 mj−1−1∑ mj=0 ( mj−1 mj ) (−T−1)∂ mj−1−mjT ∂k mj−1−mj i ∂mm0T−1 ∂k mm0 i = m0−1∏ j=1 mj−1−1∑ mj=0 ( mj−1 mj ) (−T−1)∂ mj−1−mjT ∂k mj−1−mj i × mm0−1−1∑ mm0=0 ( mm0−1 mm0 ) (−T−1)∂ mm0−1−mm0T ∂k mm0−1−mm0 i ∂mm0T−1 ∂k mm0 i = (−1)m0 m0−1∏ j=1 mj−1−1∑ mj=0 ( mj−1 mj ) T−1 ∂mj−1−mjT ∂k mj−1−mj i T−1∂mm0−1T ∂k mm0−1 i T−1. (4.5.2) Step 2 In view of (4.5.2), it is not difficult to see that ∂ mH−1k ∂km2 is given by a finite linear combination of terms of the form m∏ j=1 H−1k ∂njHk ∂k nj 2 H−1k , (4.5.3) where ∑m j=1 nj = m. Thus, when we compute ∂n ∂kn1 ∂mH−1k ∂km2 , the derivative ∂ n ∂kn1 acts either on H−1k or ∂njHk ∂k nj 2 . However, since ( ∂Hk ∂k2 ) b,c = 2(k2 + b2)δb,c− 2Â2(b− c), we have ∂n∂kn1 ∂njHk ∂k nj 2 = 0 if nj ≥ 1 and ∂n∂kn1 ∂njHk ∂k nj 2 = ∂ nHk ∂kn1 if nj = 0. Similarly, using again (4.5.2) one can see that ∂nH−1k ∂kn1 is given by a finite linear combination of terms of the form (4.5.3), with m and k2 replaced by n and k1, respectively, and ∑n j=1 nj = n. Therefore, combining all this we conclude that ∂ n+mH−1k ∂kn1 ∂k m 2 is given by a finite linear combination of terms of the formn+m∏ j=1 ∆−1k R −1 G′G′ ∂njHk ∂k nj ij ∆−1k R−1G′G′ , (4.5.4) where ∑n+m j=1 njδ2,ij = m and ∑n+m j=1 njδ1,ij = n, that is, where the sum of nj for which ij = 2 is equal to m, and the sum of nj for which ij = 1 is equal to n. Step 3 The first step in bounding (4.5.4) is to estimate∥∥∥∥∥∂njHk∂knjij ∆−1k piG′ ∥∥∥∥∥ . A simple calculation shows that ( ∂njHk ∂k nj ij ∆−1k ) b,c = 1 Nc(k) × 2(kij + bij )δb,c + 2Âij (b− c) if nj = 1, 2δb,c if nj = 2, 0 if nj ≥ 3. 90 Furthermore, by Proposition 4.2.1, 1 |Nb(k)| ≤ 1 ε|v| for all b ∈ G′, while by Proposition 3.3.1 we have 1 |Nb(k)| ≤ 2 Λ|v| (4.5.5) and |ki + bi| ≤ |ui + bi|+ |vi| ≤ |v|+ |u+ b| ≤ 2 Λ |Nb(k)| for all b ∈ G′ if G = {0, d}, and for all b ∈ G′ \ {b̃} if G = {0}. Furthermore, |b̃| ≤ Λ + |u|+ |v| < Λ + 3|v|, (4.5.6) since |u| < 2|v| because k ∈ T0 (see (4.2.2)). Now, let 1B(x) be the characteristic function of the set B. Then, using the above estimates, sup c∈G′ ∑ b∈G′ ∣∣∣(∂njHk ∂k nj ij ∆−1k piG′ ) b,c ∣∣∣ ≤ sup c∈G′ ∑ b∈G′ [ 2|kij + bij |δnj ,1 + 2δnj ,2 |Nb(k)| δb,c + 2|Âij (b− c)| |Nb(k)| δnj ,1 ] ≤ sup c∈G′ [ 2|kij + b̃ij |+ 2 |Nb̃(k)| δb̃,c + 2|Âij (b̃− c)| |Nb̃(k)| ] 1G′(b̃) + sup c∈G′ ∑ b∈G′\{b̃} [ 2|kij + bij |+ 2 |Nb(k)| δb,c + 2|Âij (b− c)| |Nb(k)| ] ≤ 2|kij + b̃ij |+ 2 + 2‖Â‖l1 ε|v| 1G′(b̃) + supc∈G′ ∑ b∈G′\{b̃} [[ 4 Λ + 2 |Nb(k)| ] δb,c + 2|Âij (b− c)| |Nb(k)| ] ≤ 2 ε|v|(2(|u|+ |v|+ |b̃|) + 2 + 2‖Â‖l1)1G′(b̃) + 4 Λ + 4 Λ|v| + 4 Λ|v|‖Â‖l1 ≤ 2 ε|v|(12|v|+ 2Λ + 2 + 2‖Â‖l1)1G′(b̃) + 4 Λ + 4 Λ|v| + 4 Λ|v|‖Â‖l1 (recall that |v| > 1) ≤ 1G′(b̃) ε−1CΛ,A + CΛ,A. 91 Similarly, sup b∈G′ ∑ c∈G′ ∣∣∣(∂njHk ∂k nj ij ∆−1k piG′ ) b,c ∣∣∣ ≤ sup b∈G′ ∑ c∈G′ [ 2|kij + bij |δnj ,1 + 2δnj ,2 |Nb(k)| δb,c + 2|Âij (b− c)| |Nb(k)| δnj ,1 ] ≤ sup b∈G′ [ 2|kij + bij |+ 2 |Nb(k)| δb,b̃ + 2|Âij (b− b̃)| |Nb(k)| ] 1G′(b̃) + sup b∈G′ ∑ c∈G′\{b̃} [ 2|kij + bij |+ 2 |Nb(k)| δb,c + 2|Âij (b− c)| |Nb(k)| ] ≤ 2|kij + b̃ij |+ 2 + 2‖Â‖l1 ε|v| 1G′(b̃) + supb∈G′ ∑ c∈G′\{b̃} [[ 4 Λ + 2 |Nb(k)| ] δb,c + 2|Âij (b− c)| |Nb(k)| ] ≤ 2 ε|v|(2(|u|+ |v|+ |b̃|) + 2 + 2‖Â‖l1)1G′(b̃) + 4 Λ + 4 Λ|v| + 4 Λ|v|‖Â‖l1 ≤ 1G′(b̃) ε−1CΛ,A + CΛ,A. Hence, by Proposition 4.1.2,∥∥∥∥∥∂njHk∂knjij ∆−1k piG′ ∥∥∥∥∥ ≤ 1G′(b̃) ε−1CΛ,A + CΛ,A. Step 4 By a similar (and much simpler) calculation (using Proposition 4.1.2) we get ‖FGG′‖ ≤ ‖f‖l1 , ‖GGG′‖ ≤ ‖g‖l1 , ‖∆−1k piG′‖ ≤ 1G′(b̃) 1 ε|v| + (1− 1G′(b̃)) 2 Λ|v| . (4.5.7) From Lemma 4.1.4 we have ‖(RG′G′)−1‖ ≤ 18. Thus, the operator norm of (4.5.4) is bounded by∥∥∥∥∥∥ n+m∏ j=1 ∆−1k R −1 G′G′ ∂njHk ∂k nj ij ∆−1k R−1G′G′ ∥∥∥∥∥∥ ≤ ‖∆−1k ‖ ‖R−1G′G′‖ n+m∏ j=1 ∥∥∥∥∥∂njHk∂knjij ∆−1k piG′ ∥∥∥∥∥ ‖R−1G′G′‖ , which is bounded either by 1 ε|v| 18 n+m∏ j=1 (ε−1CΛ,A + CΛ,A) 18 ≤ ε−(n+m+1)CΛ,A,n,m 1|v| if G = {0}, or by 1 Λ|v| 18 n+m∏ j=1 CΛ,A 18 ‖g‖l1 ≤ CΛ,A,n,m 1|v| 92 if G = {0, d}. Therefore,∥∥∥∥∥∂n+mH−1k∂kn1 ∂km2 ∥∥∥∥∥ ≤ ∑ finite sum where # of terms depend on n and m C ′ |v| ≤ Cn,m C ′ |v| ≤ C |v| , (4.5.8) with C = Cε,Λ,A,n,m if G = {0} or C = CΛ,A,n,m if G = {0, d}. Finally, recalling (4.5.1) and (4.5.7) we have∣∣∣∣ ∂n+m∂kn1 ∂km2 ΦG(k) ∣∣∣∣ = ∣∣∣∣∣FGG′ ∂n+mH−1k∂kn1 ∂km2 GG′G ∣∣∣∣∣ ≤ ‖FGG′‖ ∥∥∥∥∥∂n+mH−1k∂kn1 ∂km2 ∥∥∥∥∥ ‖GG′G‖ ≤ ‖f‖l1 C ′|v|‖g‖l1 ≤ C|v| , where C = Cε,Λ,A,n,m,f,g if G = {0} or C = CΛ,A,n,m,f,g if G = {0, d}. This is the desired inequality. The proof of the lemma is complete. Proof of Lemma 4.5.2 Let R+ be the set of non-negative real numbers and let σ be a real-valued function on R+ such that: (i) σ(t) ≥ 1 for all t ∈ R+ with σ(0) = 1; (ii) σ(s)σ(t) ≥ σ(s+ t) for all s, t ∈ R+; (iii) σ increases monotonically. For example, for any β ≥ 0 the functions t 7→ eβt and t 7→ (1 + t)β satisfy these properties. Now, let T be a linear operator from L2C to L 2 B with B,C ⊂ Γ#, (or a matrix T = [Tb,c] with b ∈ B and c ∈ C), and consider the σ-norm ‖T‖σ := max { sup b∈B ∑ c∈C |Tb,c|σ(|b− c|), sup c∈C ∑ b∈B |Tb,c|σ(|b− c|) } . In §4.9 we prove that this norm has the following properties. Proposition 4.5.4 (Properties of ‖ · ‖σ). Let S and T be linear operators from L2C to L2B with B,C ⊂ Γ#. Then: (a) ‖T‖ ≤ ‖T‖σ≡1 ≤ ‖T‖σ; (b) If B = C, then ‖S T‖σ ≤ ‖S‖σ‖T‖σ; 93 (c) If B = C, then ‖(I + T )−1‖σ ≤ (1− ‖T‖σ)−1 if ‖T‖σ < 1; (d) |Tb,c| ≤ 1σ(|b−c|)‖T‖σ for all b ∈ B and all c ∈ C. Using these properties we prove Lemma 4.5.2. Proof of Lemma 4.5.2. We follow the same notation as above. First observe that, similarly as in the last proof we can write Φd′,d′′(k) = F{d′}G′∆−1k R−1G′G′GG′{d′′} = F{d′}G′H−1k GG′{d′′}. Now, let σ(|b|) = (1 + |b|)β, and observe that there is a positive constant Cβ such that σ(|b|) ≤ Cβ(1 + |b|β) for all b ∈ Γ#. Then, it is easy to see that ‖F{d′}G′‖σ = ‖f‖σ ≤ Cβ‖(1 + |b|β)f(b)‖l1 and ‖GG′{d′′}‖σ = ‖g‖σ ≤ Cβ‖(1 + |b|β)g(b)‖l1 . Furthermore, by (4.4.36) and Proposition 4.1.2, ‖R−1G′G′‖σ = ‖(I + TG′G′)−1‖σ ≤ ∞∑ j=0 ‖TG′G‖jσ < 18, (4.5.9) and since for diagonal operators the σ-norm and the operator norm agree, from (4.5.7) we have ‖∆−1k piG′‖σ ≤ 2 Λ|v| . Hence, in view of Proposition 4.5.4(b) and Proposition 4.4.4(ii), |Φd′,d′′(k)| ≤ ‖F{d′}G′∆−1k R−1G′G′GG′{d′′}‖ ≤ Cβ,f,g,Λ,A,m,n 1 |d| , and by repeating the proof of Lemma 4.5.1 with the operator norm replaced by the σ-norm we obtain ∥∥∥∥ ∂n+m∂kn1 ∂km2 Φd′,d′′(k) ∥∥∥∥ σ ≤ Cβ,f,g,Λ,A,m,n 1|d| . Therefore, by Proposition 4.5.4(d), for any integers n and m with n+m ≥ 0,∣∣∣∣ ∂n+m∂kn1 ∂km2 Φd′,d′′(k) ∣∣∣∣ ≤ 11 + |d′ − d′′|β ∥∥∥∥ ∂n+m∂kn1 ∂km2 Φd′,d′′(k) ∥∥∥∥ σ ≤ Cβ,f,g,Λ,A,m,n 1|d|1+β . This is the desired inequality. 94 Proof of Lemma 4.5.3 Define the operator M (j) : L2G′3 → L 2 G′3 as M (j) := S if j = 1, W if j = 2, Z if j = 3, where S, W and Z are given by (4.4.23). In order to prove Lemma 4.5.3 we first prove the following proposition. Proposition 4.5.5. Assume the same hypotheses of Lemma 4.5.3. Then, for any integers n and m with n+m ≥ 1 and for 1 ≤ j ≤ 3,∥∥∥∥ ∂n+m∂kn1 ∂km2 ∆−1k M (j) ∥∥∥∥ ≤ Cj(2|zµ,d′(k)| −R)j , where C1 = C1;Λ,A,n,m and Cj = Cj;Λ,A,q,n,m for 2 ≤ j ≤ 3 are constants. Furthermore, C1;Λ,A,1,0 ≤ 13Λ2 , C1;Λ,A,0,1 ≤ 13 Λ2 and C1;Λ,A,1,1 ≤ 65Λ3 . Proof. Step 0 To simplify the notation write w = wµ,d′ , z = zµ,d′ and |z|R = 2|z| −R. First observe that, for any analytic function of the form h(k) = h̃(w(k), z(k)), we have ∂ ∂k1 h = ( ∂w ∂k1 ∂ ∂w + ∂z ∂k1 ∂ ∂z ) h̃ = ( ∂ ∂w + ∂ ∂z ) h̃, ∂ ∂k2 h = ( ∂w ∂k2 ∂ ∂w + ∂z ∂k2 ∂ ∂z ) h̃ = i(−1)ν ( ∂ ∂w − ∂ ∂z ) h̃. Thus,∥∥∥∥ ∂n+m∂kn1 ∂km2 ∆−1k M (j) ∥∥∥∥ = ∥∥∥∥∥∥(i(−1)ν)m m∑ p=0 n∑ r=0 ( m p )( n r ) (−1)m−p ∂ n−r+m−p ∂zn−r+m−p ∂r+p ∂wr+p ∆−1k M (j) ∥∥∥∥∥∥ ≤ 2n+m sup p≤r sup r≤n ∥∥∥∥ ∂n−r+m−p∂zn−r+m−p ∂r+p∂wr+p∆−1k M (j) ∥∥∥∥ . Now, by the Leibniz rule,∥∥∥∥ ∂n∂zn ∂m∂wm∆−1k M (j) ∥∥∥∥ = ∥∥∥∥∥∥ m∑ p=0 n∑ r=0 ( m p )( n r ) ∂n−r+m−p∆−1k ∂zn−r∂wm−p ∂r+pM (j) ∂zr∂wp ∥∥∥∥∥∥ ≤ 2n+m sup p≤m sup r≤n ∥∥∥∥∥∂n−r+m−p∆−1k∂zn−r∂wm−p ∥∥∥∥∥ ∥∥∥∥∥∂r+pM (j)∂zr∂wp ∥∥∥∥∥. 95 Furthermore, we shall prove below that sup p≤m sup r≤n ∥∥∥∥∥∂n−r+m−p∆−1k∂zn−r∂wm−p ∥∥∥∥∥ ∥∥∥∥∥∂r+pM (j)∂zr∂wp ∥∥∥∥∥ ≤ Cj,n,m|z|n+jR , (4.5.10) with constants C1,n,m = C1,n,m;Λ,A and Cj,n,m = Cj,n,m;Λ,A,q for 2 ≤ j ≤ 3. Hence,∥∥∥∥ ∂n∂zn ∂m∂wm∆−1k M (j) ∥∥∥∥ ≤ 2n+mCj,n,m|z|n+jR . Therefore, being careful with the indices,∥∥∥∥ ∂n+m∂kn1 ∂km2 ∆−1k M (j) ∥∥∥∥ ≤ 2n+m sup p≤m sup r≤n 2n−r+m−p+r+p Cj,n−r+m−p,r+p |z|n−r+m−p+jR ≤ Cj|z|jR , where C1 = C1;Λ,A,n,m and Cj = Cj;Λ,A,q,n,m for 2 ≤ j ≤ 3. This is the desired inequality. We are left to prove (4.5.10) and estimate the constants C1;Λ,A,i,j for i, j ∈ {0, 1} to finish the proof of the proposition. Step 1 The first step for obtaining (4.5.10) is to estimate∥∥∥∥∥∂r+p∆−1k∂zr∂wp piG′3 ∥∥∥∥∥ . Observe that∣∣∣∣∣∣ ( ∂r+p∆−1k ∂zr∂wp ) b,c ∣∣∣∣∣∣ = ∣∣∣∣∣∂r+p(∆−1k )b,c∂zr∂wp ∣∣∣∣∣ = ∣∣∣∣ ∂p∂wp 1w − 2iθµ′(b− d′) ∂ r ∂zr δb,c z − 2iθµ(b− d′) ∣∣∣∣ = ∣∣∣∣ (−1)p p!(w − 2iθµ′(b− d′))p+1 (−1) r r! δb,c (z − 2iθµ(b− d′))r+1 ∣∣∣∣ ≤ p! r! δb,c|w − 2iθµ′(b− d′)|p+1|z − 2iθµ(b− d′)|r+1 , and recall from (4.4.17) and (4.4.18) that, for all b ∈ G′3, 1 |z − 2iθµ(b− d′)| ≤ 2 |z|R and 1 |w − 2iθµ′(b− d′)| ≤ 1 Λ . (4.5.11) Then, ∣∣∣∣∣∣ ( ∂r+p∆−1k ∂zr∂wp ) b,c ∣∣∣∣∣∣ ≤ p! r! 2 r+1 δb,c Λp+1|z|r+1R , and consequently, sup b∈G′3 ∑ c∈G′3 + sup c∈G′3 ∑ b∈G′3 ∣∣∣∣∣∣ ( ∂r+p∆−1k ∂zr∂wp ) b,c ∣∣∣∣∣∣ ≤ p! r! 2 r+1 Λp+1|z|r+1R sup b∈G′3 ∑ c∈G′3 + sup c∈G′3 ∑ b∈G′3 δb,c = p! r! 2r+2 Λp+1|z|r+1R . 96 Therefore, by Proposition 4.1.2,∥∥∥∥∥∂r+p∆−1k∂zr∂wp piG′3 ∥∥∥∥∥ ≤ p! r! 2r+2Λp+1 1|z|r+1R . (4.5.12) Step 2 We now estimate the second factor in (4.5.10). Let us first consider the case j = 1, that is, M (1) = S. Since S = (I − Y33)−1, the operator S is clearly invertible. Thus, by applying (4.5.2) with T = S−1, one can see that ∂ pS ∂wp is given by a finite linear combination of terms of the form p∏ j=1 S ∂njS−1 ∂wnj S, (4.5.13) where ∑p j=1 nj = p. Hence, when we compute ∂r ∂zr ∂pS ∂wp , the derivative ∂r ∂zr acts either on S or ∂ njS−1 ∂wnj . Similarly, using again (4.5.2) with T = S−1, one can see that ∂ rS ∂zr is given by a finite linear combination of terms of the form (4.5.13), with p and w replaced by r and z, respectively, and ∑r j=1mj = r. Thus, we conclude that ∂r+pS ∂zr∂wp is given by a finite linear combination of terms of the form r+p∏ j=1 S ∂mj+njS−1 ∂zmj∂wnj S, (4.5.14) where ∑r+p j=1 mj = r and ∑r+p j=1 nj = p. Indeed, observe that the general form of the terms (4.5.14) follows directly from (4.5.2) because that identity is also valid for mixed derivatives. Since S = (I − Y33)−1 with ‖Y33‖ < 1/14 and Yb,c = −2iθµ′(Â(b− c)) z (w − 2iθµ′(c− d′))(z − 2iθµ(c− d′)) , (4.5.15) we have ‖S‖ = ‖(I − Y33)−1‖ ≤ 11− ‖Y33‖ ≤ 14 13 (4.5.16) and ∣∣∣∣∣ ( ∂j+l ∂zj∂wl S−1 ) b,c ∣∣∣∣∣ = ∣∣∣∣∣ ( ∂j+l ∂zj∂wl (I − Y33) ) b,c ∣∣∣∣∣ = ∣∣∣∣ ∂j+l∂zj∂wlYb,c ∣∣∣∣ = ∣∣∣∣∣ ∂j∂zj −2iθµ′(Â(b− c)) zz − 2iθµ(c− d′) ∂ l ∂wl 1 w − 2iθµ′(c− d′) ∣∣∣∣∣ . Furthermore, ∂j ∂zj −2iθµ′(Â(b− c)) z z − 2iθµ(c− d′) = (−1)j−1j! 2iθµ′(Â(b− c)) 2iθν(c− d′) (z − 2iθν(c− d′))j+1 for j ≥ 1, ∂l ∂wl 1 w − 2iθµ′(c− d′) = (−1)l l! (w − 2iθµ′(c− d′))l+1 for l ≥ 0. 97 Recall from (4.4.18) and (4.4.20) that, for all c ∈ G′, |c− d′| |w − 2iθµ′(c− d′)| ≤ |c− d′| |c− d′| − ε ≤ 2. (4.5.17) Then, using this and (4.5.11), for j ≥ 1 and l ≥ 0,∣∣∣∣∣ ( ∂j+l ∂zj∂wl S−1 ) b,c ∣∣∣∣∣ ≤ j! l! |Â(b− c)||z − 2iθµ(c− d′)|j+1|w − 2iθµ′(c− d′)|l |c− d ′| |w − 2iθµ′(c− d′)| ≤ 2 j+2j! l! |Â(b− c)| Λl|z|j+1R , (4.5.18) while for j = 0 and l ≥ 0,∣∣∣∣∣ ( ∂j+l ∂zj∂wl S−1 ) b,c ∣∣∣∣∣ ≤ l! |Â(b− c)| |z||z − 2iθµ(c− d′)| |w − 2iθµ′(c− d′)|l+1 ≤ 2 l! |Â(b− c)|Λl+1 . (4.5.19) Consequently, sup b∈G′3 ∑ c∈G′3 + sup c∈G′3 ∑ b∈G′3 ∣∣∣∣∣ ( ∂j+l ∂zj∂wl S−1 ) b,c ∣∣∣∣∣ ≤ ( 1− δ0,j + |z|R2Λ δ0,j ) 2j+2j! l! Λl|z|j+1R sup b∈G′3 ∑ c∈G′3 + sup c∈G′3 ∑ b∈G′3 |Â(b− c)| ≤ ( 1− δ0,j + |z|R2Λ δ0,j ) 2j+3j! l! Λl|z|j+1R ‖Â‖l1 . Therefore, by Proposition (4.1.2),∥∥∥∥ ∂j+l∂zj∂wlS−1 ∥∥∥∥ ≤ (1− δ0,j + |z|R2Λ δ0,j ) 2j+3j! l! Λl|z|j+1R ‖Â‖l1 . (4.5.20) Thus, for r ≥ 1, in view of (4.5.14) where ∑r+pj=1 mj = r,∥∥∥∥ ∂r+p∂zr∂wpS ∥∥∥∥ ≤ Cr,p r+p∏ j=1 ‖S‖ ∥∥∥∥ ∂mj+nj∂zmj∂wnj S−1 ∥∥∥∥ ‖S‖ ≤ Cr,p r+p∏ j=1 CΛ,A 2mj+3mj !nj ! Λnj ‖Â‖l1 CΛ,A r+p∏ j=1 ( 1− δ0,mj + |z|R 2Λ δ0,mj ) 1 |z|mj+1R ≤ CΛ,A,r,p 1|z|r+1R , since mj ≥ 1 for at least one 1 ≤ j ≤ r + p. Similarly, if r = 0 then∥∥∥∥ ∂r+p∂zr∂wpS ∥∥∥∥ ≤ CΛ,A,r,p. 98 Hence, in view of (4.5.12), sup p≤m sup r≤n ∥∥∥∥∥∂n−r+m−p∆−1k∂zn−r∂wm−p ∥∥∥∥∥ ∥∥∥∥∥∂r+pM (1)∂zr∂wp ∥∥∥∥∥ ≤ sup p≤m sup r≤n (m− p)! (n− r)! 2n−r+2 Λm−p+1|z|n−r+1R CΛ,A,r,p‖Â‖l1 ( 1− δ0,r + |z|R2Λ δ0,r ) 1 |z|r+1R ≤ CΛ,A,n,m 1|z|n+1R . This proves (4.5.10) for j = 1. Step 3 We now estimate the constant C1;Λ,A,i,j for i, j ∈ {0, 1}. First observe that∣∣∣∣ ∂w∂kj ∣∣∣∣ = |δ1,j + i(−1)νδ2,j | = 1 and ∣∣∣∣ ∂z∂kj ∣∣∣∣ = |δ1,j − i(−1)νδ2,j | = 1. Thus, in view of (4.5.16) and (4.5.20), since |z| ≥ |v| > R ≥ 2Λ,∥∥∥∥ ∂S∂kj ∥∥∥∥ = ∥∥∥∥−S∂S−1∂kj S ∥∥∥∥ = ∥∥∥∥−S ( ∂w∂kj ∂S −1 ∂w + ∂z ∂kj ∂S−1 ∂z ) S ∥∥∥∥ ≤ ‖S‖2 (∥∥∥∥∂S−1∂w ∥∥∥∥+ ∥∥∥∥∂S−1∂z ∥∥∥∥) ≤ (32 )2(24‖Â‖l1 |z|2R + 22‖Â‖l1 Λ2 ) ≤ ( 3 2 )2 8‖Â‖l1 Λ2 = 18‖Â‖l1 Λ2 . Similarly, ∂2S ∂ki∂kj = − ∂S ∂ki ( ∂w ∂kj ∂S−1 ∂w + ∂z ∂kj ∂S−1 ∂z ) S − S ( ∂w ∂kj ∂S−1 ∂w + ∂z ∂kj ∂S−1 ∂z ) ∂S ∂ki − S ( ∂w ∂kj ( ∂w ∂ki ∂2S−1 ∂w2 + ∂z ∂ki ∂2S−1 ∂z∂w ) + ∂z ∂kj ( ∂w ∂ki ∂2S−1 ∂w∂z + ∂z ∂ki ∂2S−1 ∂z2 )) S, so that, using the above inequality as well,∥∥∥∥ ∂2S∂ki∂kj ∥∥∥∥ ≤ 2‖S‖∥∥∥∥ ∂S∂ki ∥∥∥∥(∥∥∥∥∂S−1∂w ∥∥∥∥+ ∥∥∥∥∂S−1∂z ∥∥∥∥) + ‖S‖2 (∥∥∥∥∂2S−1∂w2 ∥∥∥∥+ 2 ∥∥∥∥∂2S−1∂z∂w ∥∥∥∥+ ∥∥∥∥∂2S−1∂z2 ∥∥∥∥) ≤ 2 3 2 18‖Â‖l1 Λ2 8‖Â‖l1 Λ2 + ( 3 2 )2(23‖Â‖l1 Λ3 + 25‖Â‖l1 Λ|z|2R + 26‖Â‖l1 |z|3R ) ≤ 432 Λ4 ‖Â‖2l1 + 54 Λ3 ‖Â‖l1 ≤ 55‖Â‖l1 Λ3 ( 8‖Â‖l1 Λ + 1 ) . Furthermore, by (4.5.12),∥∥∥∥∥∂∆−1k∂kj ∥∥∥∥∥ ≤ ∥∥∥∥∥∂∆−1k∂w ∥∥∥∥∥+ ∥∥∥∥∥∂∆−1k∂z ∥∥∥∥∥ ≤ 22Λ2|z|R + 2 3 Λ|z|2R ≤ 8 Λ2|z|R 99 and ∥∥∥∥∥∂2∆−1k∂ki∂kj ∥∥∥∥∥ ≤ ∥∥∥∥∥∂2∆−1k∂w2 ∥∥∥∥∥+ 2 ∥∥∥∥∥∂2∆−1k∂z∂w ∥∥∥∥∥+ ∥∥∥∥∥∂2∆−1k∂z2 ∥∥∥∥∥ ≤ 2 3 Λ3|z|R + 24 Λ2|z|2R + 26 Λ|z|3R < 5 · 23 Λ3 1 |z|R . Hence, since ‖Â‖l1 < 2ε/63 and ε < Λ/6,∥∥∥∥ ∂∂kj ∆−1k S ∥∥∥∥ ≤ ∥∥∥∥∥∂∆−1k∂kj ∥∥∥∥∥ ‖S‖+ ‖∆−1k ‖ ∥∥∥∥ ∂S∂kj ∥∥∥∥ ≤ 8Λ2|z|R 32 + 2Λ|z|R 18‖Â‖l1Λ2 ≤ 13Λ2 1|z|R and∥∥∥∥ ∂2∂ki∂kj ∆−1k S ∥∥∥∥ ≤ ∥∥∥∥∥∂2∆−1k∂ki∂kj ∥∥∥∥∥ ‖S‖+ ∥∥∥∥∥∂∆−1k∂kj ∥∥∥∥∥ ∥∥∥∥ ∂S∂ki ∥∥∥∥+ ∥∥∥∥∥∂∆−1k∂ki ∥∥∥∥∥ ∥∥∥∥ ∂S∂kj ∥∥∥∥+ ‖∆−1k ‖ ∥∥∥∥ ∂2S∂ki∂kj ∥∥∥∥ ≤ 1|z|R ( 5 · 23 Λ3 3 2 + 2 8 Λ2 18‖Â‖l1 Λ2 + 2 Λ 55‖Â‖l1 Λ3 (8‖Â‖l1 Λ + 1 )) < 65 Λ3 1 |z|R . Therefore, C1;Λ,A,1,0 ≤ 13Λ2 , C1;Λ,A,0,1 ≤ 13 Λ2 and C1;Λ,A,1,1 ≤ 65Λ3 , as was to be shown. Step 4 To prove (4.5.10) for j = 2 we need to bound∥∥∥∥∥∂r+pM (2)∂zr∂wp ∥∥∥∥∥ = ∥∥∥∥ ∂r+pW∂zr∂wp ∥∥∥∥ . Recall from (4.4.23) that W = ∞∑ j=1 Wj = ∞∑ j=1 j∑ m=1 (Y33)m−1X33(Y33)j−m, where Yb,c is given above by (4.5.15) and ‖X33‖ ≤ C/|z| < 1/3 with Xb,c = (c− d′) · Â(b− c)− q̂(b− c)− 2iθµ(Â(b− c))w (w − 2iθµ′(c− d′))(z − 2iθµ(c− d′)) . First observe that ∂r+p ∂zrwp (Y33)m−1X33(Y33)j−m is given by a sum of jr+p terms of the form ∂l1+n1Y33 ∂zl1∂wn1 · · · ∂ lm−1+nm−1Y33 ∂zlm−1∂wnm−1 ∂lm+nmX33 ∂zlm∂wnm ∂lm+1+nm+1Y33 ∂zlm+1∂wnm+1 · · · ∂ lj+njY33 ∂zlj∂wnj , 100 where there are j factors ordered as in the product (Y33)m−1X33(Y33)j−m. Furthermore, for each term in the sum we have ∑j i=1 li = r and ∑j i=1 ni = p. Thus,∥∥∥∥ ∂r+p∂zrwp W ∥∥∥∥ = ∥∥∥∥∥∥ ∞∑ j=1 ∂r+p ∂zrwp Wj ∥∥∥∥∥∥ = ∥∥∥∥∥∥ ∞∑ j=1 j∑ m=1 ∂r+p ∂zrwp (Y33)m−1X33(Y33)j−m ∥∥∥∥∥∥ (4.5.21) ≤ ∞∑ j=1 j∑ m=1 ∥∥∥∥ ∂r+p∂zrwp (Y33)m−1X33(Y33)j−m ∥∥∥∥ ≤ ∞∑ j=1 jr+p j∑ m=1 sup I ∥∥∥∥∂l1+n1Y33∂zl1∂wn1 · · · ∂lm+nmX33∂zlm∂wnm · · · ∂lj+njY33∂zlj∂wnj ∥∥∥∥ ≤ ∞∑ j=1 jr+p j∑ m=1 sup I ∥∥∥∥∂l1+n1Y33∂zl1∂wn1 ∥∥∥∥ · · · ∥∥∥∥∂lm+nmX33∂zlm∂wnm ∥∥∥∥ · · · ∥∥∥∥∂lj+njY33∂zlj∂wnj ∥∥∥∥ , (4.5.22) where I := { (li, ni) ∣∣∣∣∣ li ≤ r and ni ≤ p for 1 ≤ i ≤ j with j∑ i=1 li = r and j∑ i=1 ni = p } . (4.5.23) Note, we can differentiate the series (4.5.21) term-by-term because the sum ∑∞ j=1Wj con- verges uniformly and the sum ∑j m=1 is finite. We next estimate the factors in (4.5.22). Combining (4.5.18) and (4.5.19) we have∣∣∣∣ ∂li+ni∂zli∂wni Yb,c ∣∣∣∣ ≤ (1− δ0,li + |z|R2Λ δ0,li ) 2li+2li!ni! Λni |z|li+1R |Â(b− c)|. (4.5.24) Furthermore, using (4.5.11) and (4.5.17),∣∣∣∣ ∂li+ni∂zli∂wniXb,c ∣∣∣∣ = ∣∣∣∣∣ ∂li∂zli 1z − 2iθµ(c− d′) ∂ ni ∂wni (c− d′) · Â(b− c)− q̂(b− c)− 2iθµ(Â(b− c))w w − 2iθµ′(c− d′) ∣∣∣∣∣ = ∣∣∣∣∣(−1)li li! (−1)ni ni! (2θµ(Â(b− c)) 2θµ′(c− d′)− (c− d′) · Â(b− c)− q̂(b− c))(z − 2iθµ(c− d′))li+1(w − 2iθµ′(c− d′))ni+1 ∣∣∣∣∣ ≤ li!ni! (2|Â(b− c)| |c− d ′|+ |q̂(b− c)|) |z − 2iθµ(c− d′)|li+1|w − 2iθµ′(c− d′)|ni+1 ≤ 2 li+1li!ni! Λni |z|li+1R 2|Â(b− c)| |c− d′|+ |q̂(b− c)| |w − 2iθµ′(c− d′)| ≤ 2 li+1li!ni! Λni |z|li+1R ( 4|Â(b− c)|+ 1 Λ |q̂(b− c)| ) . (4.5.25) 101 Hence, sup b∈G′3 ∑ c∈G′3 + sup c∈G′3 ∑ b∈G′3 ∣∣∣∣ ∂li+ni∂zli∂wni Yb,c ∣∣∣∣ ≤ ( 1− δ0,li + |z|R 2Λ δ0,li ) 2li+2li!ni! Λni |z|li+1R sup b∈G′3 ∑ c∈G′3 + sup c∈G′3 ∑ b∈G′3 |Â(b− c)| ≤ ( 1− δ0,li + |z|R 2Λ δ0,li ) 2li+3li!ni! Λni |z|li+1R ‖Â‖l1 and sup b∈G′3 ∑ c∈G′3 + sup c∈G′3 ∑ b∈G′3 ∣∣∣∣ ∂li+ni∂zli∂wniXb,c ∣∣∣∣ ≤ 2 li+1li!ni! Λni |z|li+1R sup b∈G′1 ∑ c∈G′1 + sup c∈G′1 ∑ b∈G′1 (4|Â(b− c)|+ |q̂(b− c)| Λ ) ≤ 2 li+2li!ni! Λni |z|li+1R ( 4‖Â‖l1 + ‖q̂‖l1 Λ ) . Thus, by Proposition (4.1.2), since |z| ≥ |v| > R ≥ 2Λ,∥∥∥∥ ∂li+ni∂zli∂wni Y33 ∥∥∥∥ ≤ (1− δ0,li + |z|R2Λ δ0,li ) 2li+3li!ni! Λni |z|li+1R ‖Â‖l1 ≤ ( 1 |z|R + 1 2Λ ) 2li+3li!ni! Λni |z|liR ‖Â‖l1 ≤ 2li+3li!ni! Λni+1|z|liR ‖Â‖l1 (4.5.26) and∥∥∥∥ ∂li+ni∂zli∂wniX33 ∥∥∥∥ ≤ 2li+2li!ni!Λni |z|li+1R ( 4‖Â‖l1 + ‖q̂‖l1 Λ ) = ( 2Λ + ‖q̂‖l1 2‖Â‖l1 ) 1 |z|R 2li+3li!ni! Λni+1|z|liR ‖Â‖l1 . (4.5.27) Applying these estimates to (4.5.22) and recalling that ∑j i=1 li = r and ∑j i=1 ni = p we have∥∥∥∥ ∂r+p∂zrwpW ∥∥∥∥ ≤ ∞∑ j=1 jr+p j∑ m=1 sup I ∥∥∥∥∂l1+n1Y33∂zl1∂wn1 ∥∥∥∥ · · · ∥∥∥∥∂lm+nmX33∂zlm∂wnm ∥∥∥∥ · · · ∥∥∥∥∂lj+njY33∂zlj∂wnj ∥∥∥∥ ≤ ∞∑ j=1 jr+p j∑ m=1 sup I {( 2Λ + ‖q̂‖l1 2‖Â‖l1 ) 1 |z|R j∏ i=1 2li+3li!ni! Λni+1|z|liR ‖Â‖l1 } = ( 2Λ + ‖q̂‖l1 2‖Â‖l1 ) 1 |z|R 2r Λp|z|rR ∞∑ j=1 jr+p ( 8‖Â‖l1 Λ )j sup I { j∏ i=1 li! j∏ m=1 nm! } j∑ m=1 1 ≤ ( 2Λ + ‖q̂‖l1 2‖Â‖l1 ) 2rr!p! Λp|z|r+1R ∞∑ j=1 jr+p+1 ( 1 21 )j ≤ C ′Λ,A,q,r,p 1 |z|r+1R . 102 This is the inequality we needed to prove (4.5.10) for j = 2. In fact, using (4.5.12) we obtain sup p≤m sup r≤n ∥∥∥∥∥∂n−r+m−p∆−1k∂zn−r∂wm−p ∥∥∥∥∥ ∥∥∥∥∥∂r+pM (2)∂zr∂wp ∥∥∥∥∥ ≤ supp≤m supr≤n (m− p)! (n− r)! 2 n−r+2 Λm−p+1|z|n−r+1R C ′Λ,A,q,r,p |z|r+1R ≤ CΛ,A,q,m,n 1|z|n+2R . Step 5 To prove (4.5.10) for j = 3 we need to estimate∥∥∥∥∥∂r+pM (3)∂zr∂wp ∥∥∥∥∥ = ∥∥∥∥ ∂r+pZ∂zr∂wp ∥∥∥∥ , where Z = ∞∑ j=2 Zj = ∞∑ j=2 (X33 + Y33)j −Wj − Y j33. First observe that ∂r+p ∂zr∂wp Zj = ∂r+p ∂zr∂wp ((X33 + Y33)j −Wj − Y j33) is given by a sum of (2j − j − 1) · jr+p terms of the form ∂l1+n1Y33 ∂zl1∂wn1 · · · ∂ lm+nmX33 ∂zlm∂wnm · · · ∂ lj+njY33 ∂zlj∂wnj︸ ︷︷ ︸ j factors , (4.5.28) where there are j − 2 factors involving X33 or Y33 and two factors containing X33. Further- more, for each term in the sum we have ∑j i=1 li = r and ∑j i=1 ni = p. Thus,∥∥∥∥ ∂r+p∂zr∂wpZj ∥∥∥∥ ≤ (2j − j − 1) jr+p supI ∥∥∥∥∂l1+n1Y33∂zl1∂wn1 ∥∥∥∥ · · · ∥∥∥∥∂lm+nmX33∂zlm∂wnm ∥∥∥∥ · · · ∥∥∥∥∂lj+njY33∂zlj∂wnj ∥∥∥∥ , where the set I is given above by (4.5.23). Now observe that, the estimate for the derivatives of X33 in (4.5.27) is better then the estimate for the derivatives of Y33 in (4.5.26) because the former has an extra factor CΛ,A,q/|z|R < 1. Since the product (4.5.28) has at least two factors containing X33, we can estimate any of these products by considering the worst case. This happens when there are exactly two factors involving X33. Hence, by proceeding in this way, for each j ≥ 2 we have∥∥∥∥ ∂r+p∂zr∂wpZj ∥∥∥∥ ≤ (2j − j − 1) jr+p supI ( 2Λ + ‖q̂‖l1 2‖Â‖l1 )2 1 |z|2R j∏ i=1 2li+3li!ni! Λni+1|z|liR ‖Â‖l1 ≤ 2jjr+p ( 2Λ + ‖q̂‖l1 2‖Â‖l1 )2 1 |z|2R 2rr!p! Λp|z|rR ( 8‖Â‖l1 Λ )j ≤ C ′Λ,A,q,r,p jr+p ( 2 21 )j 1 |z|r+2R , 103 since ‖A‖l1 ≤ 2ε/63 and ε < Λ/6. Thus,∥∥∥∥ ∂r+p∂zr∂wpZ ∥∥∥∥ ≤ ∞∑ j=2 ∥∥∥∥ ∂r+p∂zr∂wpZj ∥∥∥∥ ≤ C ′Λ,A,q,r,p|z|r+2R ∞∑ j=2 jr+p ( 2 21 )j ≤ CΛ,A,q,r,p|z|r+2R . Therefore, recalling (4.5.12), sup p≤m sup r≤n ∥∥∥∥∥∂n−r+m−p∆−1k∂zn−r∂wm−p ∥∥∥∥∥ ∥∥∥∥∥∂r+pM (3)∂zr∂wp ∥∥∥∥∥ ≤ supp≤m supr≤n (m− p)! (n− r)! 2 n−r+2 Λm−p+1|z|n−r+1R C ′Λ,A,q,r,p |z|r+2R ≤ CΛ,A,q,m,n 1|z|n+3R . This is the desired inequality for j = 3. The proof of the proposition is complete. We can now prove Lemma 4.5.3. We first prove it for 1 ≤ j ≤ 2 and then for j = 3 separately. Proof of Lemma 4.5.3 for 1 ≤ j ≤ 2. Define the |B| × |C| matrices FBC := [f(b− c)]b∈B,c∈C and GBC := [g(b− c)]b∈b,c∈C , and write w = wµ,d′ , z = zµ,d′ and |z|R = 2|z| −R. First observe that, for 1 ≤ j ≤ 2, the functions [ α (j) µ,d′(k) ] d′∈G = ∑ b,c∈G′1 f(d′ − b)M (j)b,c g(c− d′) (w − 2iθµ′(b− d′))(z − 2iθµ(b− d′)) d′∈G are the diagonal entries of the matrix FGG′1∆−1k M (j) GG′1G. Thus, similarly as in the proof of Lemma 4.5.1, by Proposition 4.5.5, for 1 ≤ j ≤ 2,∣∣∣∣∣ ∂n+m∂kn1 ∂km2 α(j)µ,d′(k) ∣∣∣∣∣ ≤ ‖FGG′1‖ ∥∥∥∥ ∂n∂kn1 ∂ m ∂km2 ∆−1k M (j) ∥∥∥∥ ‖GG′1G‖ ≤ ‖f‖l1‖g‖l1 ∥∥∥∥ ∂n∂kn1 ∂ m ∂km2 ∆−1k M (j) ∥∥∥∥ ≤ Cj|z|jR , where C1 = C1;Λ,A,n,m,f,g and C2 = C2;Λ,A,q,n,m,f,g are constants. Furthermore, C1;Λ,A,1,0,f,g ≤ 13Λ2 ‖f‖l1‖g‖l1 , C1;Λ,A,0,1,f,g ≤ 13 Λ2 ‖f‖l1‖g‖l1 and C1;Λ,A,1,1,f,g ≤ 65Λ3 ‖f‖l1‖g‖l1 . This proves the lemma for 1 ≤ j ≤ 2. 104 Proof of Lemma 4.5.3 for j = 3. We need to estimate ∂n+m ∂kn1 ∂k m 2 α (3) µ,d′(k) = 4∑ j=1 ∂n+m ∂kn1 ∂k m 2 Rj(k), where R1, . . . ,R4 are given by (4.4.5), (4.4.6), (4.4.34) and (4.4.26), respectively. Step 1 We begin with the terms involving R1 and R2, which are easier. We follow the same notation as above. First observe that, similarly as in the proof of Lemma 4.5.1, since ∆−1k R −1 G′G′ = H −1 k on L 2 G′ , we have∣∣∣∣ ∂n+m∂kn1 ∂km2 R1(k) ∣∣∣∣ = ∥∥∥∥∥F{d′}G′1 ∂n+mH−1k∂kn1 ∂km2 GG′2{d′} ∥∥∥∥∥ ≤ ‖F{d′}G′1‖ ∥∥∥∥∥∂n+mH−1k∂kn1 ∂km2 ∥∥∥∥∥ ‖GG′2{d′}‖,∣∣∣∣ ∂n+m∂kn1 ∂km2 R2(k) ∣∣∣∣ = ∥∥∥∥∥F{d′}G′2 ∂n+mH−1k∂kn1 ∂km2 GG′{d′} ∥∥∥∥∥ ≤ ‖F{d′}G′2‖ ∥∥∥∥∥∂n+mH−1k∂kn1 ∂km2 ∥∥∥∥∥ ‖GG′{d′}‖. Furthermore, we have already proved that (see (4.5.7) and (4.5.8)) ‖F{d′}G′1‖ ≤ ‖f‖l1 , ‖GG′{d′}‖ ≤ ‖g‖l1 and, since |z| ≤ 3|v| by Proposition 4.4.4,∥∥∥∥∥∂n+mH−1k∂kn1 ∂km2 ∥∥∥∥∥ ≤ ε−(n+m+1)CΛ,A,n,m 1|z| . Now recall that G′2 = {b ∈ G′ | |b− d′| > 14R}. Then, sup b∈{d′} ∑ c∈G′2 |f(b− c)| ≤ ∑ c∈G′2 |d′ − c|2 |d′ − c|2 |f(d ′ − c)| ≤ ‖b2f(b)‖l1 sup c∈G′2 1 |d′ − c|2 ≤ 16 R2 ‖b2f(b)‖l1 , sup c∈G′2 ∑ b∈{d′} |f(b− c)| ≤ sup c∈G′2 |d′ − c|2 |d′ − c|2 |f(d ′ − c)| ≤ ‖b2f(b)‖l1 sup c∈G′2 1 |d′ − c|2 ≤ 16 R2 ‖b2f(b)‖l1 . Hence, by Proposition (4.1.2), ‖F{d′}G′2‖ ≤ 16‖b2f(b)‖l1 1 R2 . Similarly, ‖GG′2{d′}‖ ≤ 16‖b2f(b)‖l1 1 R2 . Therefore, combining all this, for 1 ≤ j ≤ 2 we obtain∣∣∣∣ ∂n+m∂kn1 ∂km2 Rj(k) ∣∣∣∣ ≤ ε−(n+m+1)CΛ,A,n,m,f,g 1|z|R2 . 105 Step 2 Recall from (4.4.26) the expression forR4. Then, similarly as above, by applying Proposition 4.5.5 for j = 3 we find that∣∣∣∣ ∂n+m∂kn1 ∂km2 R4(k) ∣∣∣∣ ≤ ‖F{d′}G′1‖ ∥∥∥∥ ∂n+m∂kn1 ∂km2 ∆−1k Z ∥∥∥∥ ‖GG′1{d′}‖ ≤ ‖f‖l1‖g‖l1CΛ,A,q,n,m 1|z|3R . Step 3 To bound the derivatives of R3 (which is given by (4.4.34)) we need a few more estimates. Recall from (4.4.29) that W (j)43 = piG′4T j+1 G′G′piG′3 . First observe that ∂r+p ∂kr1∂k p 2 piG′1∆ −1 k T m 33T34W (j−m−1) 43 = ∂r+p ∂kr1∂k p 2 ∆−1k piG′1T m 33T34T j−m G′G′ piG′3 is given by a sum of (j + 2)r+p terms of the form ∂l1+n1∆−1k ∂kl11 ∂k n1 2 piG′1 ∂l2+n2T33 ∂kl21 ∂k n2 2 · · · ∂ lm+2+nm+2T34 ∂k lm+2 1 ∂k nm+2 2 ∂lm+3+nm+3TG′G′ ∂k lm+3 2 ∂k nm+3 2 · · · ∂ lj+2+nj+2TG′G′ ∂k lj+2 1 ∂k nj+2 2 piG′3 . Moreover, for each term in the sum we have ∑j+2 i=1 li = r and ∑j+2 i=1 ni = p. Thus,∥∥∥∥ ∂r+p∂kr1∂kp2 piG′1∆−1k Tm33T34W (j−m−1)43 ∥∥∥∥ ≤ (j + 2)r+p supI′ ∥∥∥∥∥ ( j+2∏ i=1 ∂li+niT(i) ∂kli1 ∂k ni 2 ) piG′3 ∥∥∥∥∥ , (4.5.29) where the set I ′ is given by (4.5.23) with j replaced by j + 2 and T(i) := ∆−1k piG′1 for i = 1, T33 for 2 ≤ i ≤ m+ 1, T34 for i = m+ 2, TG′G′ for m+ 3 ≤ i ≤ j + 2. (4.5.30) Step 3a The first step in bounding (4.5.29) is to estimate∥∥∥∥∥∂r+p∆−1k∂kr1∂kp2 piG′1 ∥∥∥∥∥ . We follow the same argument that we have used in the proof of Lemma 4.5.1 to bound∥∥∥∥∥∂n+mH−1k∂kn1 ∂km2 ∥∥∥∥∥ . In fact, in view of (4.5.2) one can see that ∂p∆−1k ∂kp2 = ∑ finite sum where # of terms depend on p p∏ j=1 ∆−1k ∂nj∆k ∂k nj 2 ∆−1k , (4.5.31) 106 where ∑p j=1 nj = p. Hence, when we compute ∂r ∂kr1 ∂p∆−1k ∂kp2 , the derivative ∂ r ∂kr1 acts either on ∆−1k or ∂nj∆k ∂k nj 2 . However, since ( ∂∆k ∂k2 ) b,c = 2(k2 + c2)δb,c, we have ∂ r ∂kr1 ∂nj ∂k nj 2 ∆k = 0 if nj ≥ 1 and ∂ r ∂kr1 ∂nj ∂k nj 2 ∆k = ∂ r ∂kr1 ∆k if nj = 0. Similarly, using again (4.5.2) one can see that ∂r∆−1k ∂kr1 is given by a finite sum as in (4.5.31), with p and k2 replaced by r and k1, respectively, and∑r j=1 nj = r. Thus, combining all this we conclude that ∂r+p∆−1k ∂kr1∂k p 2 = ∑ finite sum where # of terms depend on r and p r+p∏ j=1 ∆−1k ∂nj∆k ∂k nj ij ∆−1k , (4.5.32) where ∑r+p j=1 njδ2,ij = p and ∑r+p j=1 njδ1,ij = r. If we observe that ( ∂nj∆k ∂k nj ij ) b,c = 2(kij + cij )δb,c if nj = 1, 2δb,c if nj = 2, 0 if nj ≥ 3, and extract the “leading term” from the summation in (4.5.32), in a sense that will be clear below, we can rewrite (4.5.32) in terms of matrix elements as ∂r+p ∂kr1∂k p 2 1 Nc(k) = (−1)r+p(r + p)! Nc(k) [ 2(k1 + c1) Nc(k) ]r [2(k2 + c2) Nc(k) ]p + ∑ finite sum where # of terms depend on r and p (2(k1 + c1))αj (2(k2 + c2))βj Nc(k)r+p+1 , where αj + βj < r + p for every j in the summation. Recall from (4.5.5) and (4.5.6) that, for all c ∈ G′ \ {c̃}, |ki + ci| |Nc(k)| ≤ 2 Λ < 1 3ε < 7 2ε and |ki + c̃i| |Nc̃(k)| ≤ Λ + 3|v| ε|v| ≤ 7 2ε . (4.5.33) Hence,∣∣∣∣ ∂r+p∂kr1∂kp2 1Nc(k) ∣∣∣∣ ≤ (r + p)!|Nc(k)| ( 7 ε )r+p + ∑ finite sum where # of terms depend on r and p ( 7 ε )αj+βj 1 |Nc(k)|2 ≤ (r + p)!|Nc(k)| ( 7 ε )r+p + Cε,r,p 1 |Nc(k)|2 . (4.5.34) Thus, by Proposition 4.1.2, since |Nc(k)| ≥ ε|v| ≥ ε|z|/3 for all c ∈ G′, we have∥∥∥∥∥∂r+p∆−1k∂kr1∂kp2 piG′1 ∥∥∥∥∥ ≤ 7r+p(r + p)!εr+p+1 3|z| + Cε,r,p|z|2 . (4.5.35) 107 Now, let ρ1 = ρ1;ε,r,p be the constant ρ1;ε,r,p := max l1≤r n1≤p εl1+n1+1Cε,l1,n1 4(l1 + n1)! 7l1+n1 , where Cε,l1,n1 is the constant in (4.5.35). Then, for |z| > ρ1 and for any l1 ≤ r and any n1 ≤ p, ∥∥∥∥∥∂l1+n1∆−1k∂kl11 ∂kn12 piG′1 ∥∥∥∥∥ ≤ 7l1+n1(l1 + n1)!εl1+n1+1 3|z| + 7l1+n1(l1 + n1)!εl1+n1+1 4|z| = (l1 + n1)! ( 7 ε )l1+n1+1 1 |z| . (4.5.36) This is the first inequality we need to bound (4.5.29). We next estimate the other factors in that expression. Step 3b Recall from (4.4.12) that Tb,c = 1 Nc(k) (2(c+ k) · Â(b− c)− q̂(b− c)). By direct calculation we have ∂r+p Tb,c ∂kr1∂k p 2 = ( ∂r+p ∂kr1∂k p 2 1 Nc(k) ) (2(c+ k) · Â(b− c)− q̂(b− c)) + r ( ∂r−1+p ∂kr−11 ∂k p 2 1 Nc(k) ) 2Âj(b− c) + p ( ∂r+p−1 ∂kr1∂k p−1 2 1 Nc(k) ) 2Âj(b− c). Hence, using (4.5.33) and (4.5.34), since |Nc(k)| ≥ ε|v| ≥ ε|z|/3 for all c ∈ G′ and |v| > 1,∣∣∣∣∂r+p Tb,c∂kr1∂kp2 ∣∣∣∣ ≤ ( (r + p)! ( 7 ε )r+p + Cε,r,p ε|v| )( 7 ε |Â(b− c)|+ |q̂(b− c)| ε|v| ) + Cε,r,p |v| |Â(b− c)| ≤ (r + p)! ( 7 ε )r+p+1 |Â(b− c)|+ Cε,r,p|z| (|Â(b− c)|+ |q̂(b− c)|). (4.5.37) Therefore, by Proposition 4.1.2, ∥∥∥∥∂r+p TG′G′∂kr1∂kp2 ∥∥∥∥ ≤ Θr,p, (4.5.38) where Θr,p := (r + p)! ( 7 ε )r+p+1 ‖Â‖l1 + Cε,A,q,r,p 1 |z| . (4.5.39) This is the second estimate we need to bound (4.5.29). We next derive one more inequality. 108 Step 3c Set Qr,pb,c := (1 + |b− c|2) ∂r+p Tb,c ∂kr1∂k p 2 . We first prove that, for any B,C ⊂ G′, sup b∈B ∑ c∈C |Qr,pb,c | ≤ Ωr,p and sup c∈C ∑ b∈B |Qr,pb,c | ≤ Ωr,p, where Ωr,p := (r + p)! ( 7 ε )r+p+1 ‖(1 + b2)Â(b)‖l1 + Cε,A,q,r,p 1 |z| . (4.5.40) In fact, in view of (4.5.37) we have sup b∈B ∑ c∈C |Qr,pb,c | = sup b∈B ∑ c∈C (1 + |b− c|2) ∣∣∣∣∂r+pTb,c∂kr1∂kp2 ∣∣∣∣ ≤ sup b∈B ∑ c∈C (1 + |b− c|2) × [ (r + p)! ( 7 ε )r+p+1 |Â(b− c)|+ Cε,r,p|z| (|Â(b− c)|+ |q̂(b− c)|) ] ≤ (r + p)! ( 7 ε )r+p+1 ‖(1 + b2)Â(b)‖l1 + Cε,A,q,r,p 1 |z| , and similarly we estimate supc∈C ∑ b∈B |Qr,pb,c |. Now observe that, as in (4.4.37), for any integer m ≥ 0 and for any ξ0, ξ1, . . . , ξm+2 ∈ Γ#, let b = ξ0 and c = ξm+2. Then, |b− c|2 ≤ 2(m+ 2) m+2∑ i=1 |ξi−1 − ξi|2. To simplify the notation write ∂li,ni = ∂li+ni ∂kli1 ∂k ni 2 , and recall from (4.5.30) and (4.5.40) the definition of T(i) and Ωr,p. Hence, similarly as in the proof of Proposition 4.4.6, since |b− c| ≥ R/4 for all b ∈ G′1 and c ∈ G′4, 109 sup b∈G′1 ∑ c∈G′4 ∣∣∣∣∣∣ ( m+2∏ i=2 ∂li,niT(i) ) b,c ∣∣∣∣∣∣ ≤ sup b∈G′1 c∈G′4 1 1 + |b− c|2 supb∈G′1 ∑ c∈G′4 (1 + |b− c|2) ∣∣∣∣∣∣ ( m+2∏ i=2 ∂li,niT(i) ) b,c ∣∣∣∣∣∣ ≤ 2(m+ 2) 1 + 116R 2 sup b∈G′1 ∑ ξ1∈G′3 (1 + |b− ξ1|2) ∣∣∣∂l2,n2Tb,ξ1∣∣∣ × ∑ ξ2∈G′3 (1 + |ξ1 − ξ2|2) ∣∣∣∂l3,n3Tξ1,ξ2∣∣∣ · · ·∑ c∈G′4 (1 + |ξm+1 − c|2) ∣∣∣∂lm+2,nm+2Tξm+1,c∣∣∣ ≤ 2(m+ 2) 1 + 116R 2 sup b∈G′1 ∑ ξ1∈G′3 (1 + |b− ξ1|2) ∣∣∣∂l2,n2Tb,ξ1∣∣∣ sup ξ1∈G′3 ∑ ξ2∈G′3 (1 + |ξ1 − ξ2|2) ∣∣∣∂l3,n3Tξ1,ξ2∣∣∣ × sup ξm+1∈G′3 ∑ c∈G′4 (1 + |ξm+1 − c|2) ∣∣∣∂lm+2,nm+2Tξm+1,c∣∣∣ = 2(m+ 2) 1 + 116R 2 sup b∈G′1 ∑ ξ1∈G′3 |Ql2,n2b,ξ1 | · · · sup ξm+1∈G′3 ∑ c∈G′4 |Qlm+2,nm+2ξm+1,c | ≤ 2(m+ 2) 1 + 116R 2 m+2∏ i=2 Ωli,ni and sup c∈G′4 ∑ b∈G′1 ∣∣∣∣∣∣ ( m+2∏ i=2 ∂li,niT(i) ) b,c ∣∣∣∣∣∣ ≤ sup b∈G′1 c∈G′4 1 1 + |b− c|2 supc∈G′4 ∑ b∈G′1 (1 + |b− c|2) ∣∣∣∣∣∣ ( m+2∏ i=2 ∂li,niT(i) ) b,c ∣∣∣∣∣∣ ≤ 2(m+ 2) 1 + 116R 2 sup c∈G′4 ∑ ξm+1∈G′3 (1 + |ξm+1|2) ∣∣∣∂lm+2,nm+2Tξm+1,c∣∣∣ × ∑ ξm∈G′3 (1 + |ξm − ξm+1|2) ∣∣∣∂lm+1,nm+1Tξm,ξm+1∣∣∣ · · ·∑ b∈G′1 (1 + |b− ξ1|2) ∣∣∣∂l2,n2Tb,ξ1∣∣∣ ≤ 2(m+ 2) 1 + 116R 2 sup c∈G′4 ∑ ξm+1∈G′3 (1 + |ξm+1|2) ∣∣∣∂lm+2,nm+2Tξm+1,c∣∣∣ × sup ξm+1∈G′3 ∑ ξm∈G′3 (1 + |ξm − ξm+1|2) ∣∣∣∂lm+1,nm+1Tξm,ξm+1∣∣∣ × sup ξ1∈G′3 ∑ b∈G′1 (1 + |b− ξ1|2) ∣∣∣∂l2,n2Tb,ξ1∣∣∣ = 2(m+ 2) 1 + 116R 2 sup c∈G′4 ∑ ξm+1∈G′3 |Qlm+2,nm+2ξm+1,c | · · · sup ξ1∈G′3 ∑ b∈G′1 |Ql2,n2b,ξ1 | ≤ 2(m+ 2) 1 + 116R 2 m+2∏ i=2 Ωli,ni . 110 Therefore, by Proposition 4.1.2,∥∥∥∥∥piG′1 m+2∏ i=2 ∂li+niT(i) ∂kli1 ∂k ni 2 ∥∥∥∥∥ ≤ 2(m+ 2)1 + 116R2 m+2∏ i=2 Ωli,ni . We have all we need to bound (4.5.29). Step 3d From (4.5.38) and (4.5.36) it follows that∥∥∥∥∥ j+2∏ i=m+3 ∂li+niT(i) ∂kli1 ∂k ni 2 ∥∥∥∥∥ ≤ j+2∏ i=m+3 Θli,ni and ∥∥∥∥∥∂l1+n1T(1)∂kl11 ∂kn12 ∥∥∥∥∥ ≤ (r + p)! ( 7 ε )r+p+1 1 |z| . Thus, recalling (4.5.29) we get∥∥∥∥ ∂r+p∂kr1∂kp2 ∆−1k piG′1Tm33T34W (j−m−1)43 ∥∥∥∥ ≤ (j + 2)r+p supI′ ∥∥∥∥∥ ( j+2∏ i=1 ∂li+niT(i) ∂kli1 ∂k ni 2 ) piG′3 ∥∥∥∥∥ ≤ (j + 2)r+p sup I′ { 1 |z| 2(m+ 2) 1 + 116R 2 (r + p)! ( 7 ε )r+p+1 [m+2∏ i=2 Ωli,ni ] j+2∏ i=m+3 Θli,ni } ≤ (j + 2)r+p(m+ 2) C|z|R2 supI′ { (l1 + n1)! ( 7 ε )l1+n1+1 [m+2∏ i=2 Ωli,ni ] j+2∏ i=m+3 Θli,ni } , where C is an universal constant. Now, recall the definition of Θr,p and Ωr,p in (4.5.39) and (4.5.40), observe that ‖Â‖l1 < ‖(1 + b2)Â‖l1 , and let ρ2 = ρ2;ε,A,q,r,p be a sufficiently large constant such that, for |z| > ρ2 and for any li ≤ r and any ni ≤ p, Θli,ni , Ωli,ni ≤ 2(li + ni)! ( 7 ε )li+ni+1 ‖(1 + b2)Â(b)‖l1 . Then,∥∥∥∥ ∂r+p∂kr1∂kp2 ∆−1k piG′1Tm33T34W (j−m−1)43 ∥∥∥∥ ≤ (j + 2)r+p(m+ 2) C|z|R2 supI′ { (l1 + n1)! ( 7 ε )l1+n1+1 [m+2∏ i=2 Ωli,ni ] j+2∏ i=m+3 Θli,ni } ≤ (j + 2)r+p (m+ 2)C|z|R2 (2‖(1 + b 2)Â(b)‖l1)j+1 ( 7 ε )j+2 sup I′ {( 7 ε )Pj+2 i=1 (li+ni) j+2∏ i=1 (li + ni)! } (since ∑j+2 i=1 li = r, ∑j+2 i=1 ni = p and ∏j+2 i=1 (li + ni)! < (r + p)!) ≤ C(r + p)! ( 7 ε )r+p+1 (m+ 2)(j + 2)r+p ( 14 ε ‖(1 + b2)Â(b)‖l1 )j+1 1 |z|R2 ≤ Cε,r,p|z|R2 (m+ 2)(j + 2) r+p ( 4 9 )j+1 , 111 since ‖(1 + b2)Â(b)‖l1 < 2ε/63. This establishes a bound for (4.5.29). Step 4 We now apply the last inequality for deriving an estimate for the derivatives of R3 and complete the proof of the lemma for j = 3. Recall from (4.4.35) that X (j) 33 = j−1∑ m=0 Tm33T34W (j−m−1) 43 . Then,∥∥∥∥ ∂r+p∂kr1∂kp2 piG′1∆−1k X(j)33 ∥∥∥∥ ≤ j−1∑ m=0 ∥∥∥∥ ∂r+p∂kr1∂kp2 ∆−1k piG′1Tm33T34W (j−m−1)43 ∥∥∥∥ ≤ j−1∑ m=0 Cε,r,p |z|R2 (m+ 2)(j + 2) r+p ( 4 9 )j+1 ≤ Cε,r,p|z|R2 (j + 2) r+p ( 4 9 )j+1 j−1∑ m=0 (m+ 2) = Cε,r,p |z|R2 (j + 2) r+p ( 4 9 )j+1 1 2 (j2 + 3j). Thus, since G′1 ⊂ G′3,∥∥∥∥∥∥piG′1 ∂ r+p ∂kr1∂k p 2 ∆−1k ∞∑ j=1 X (j) 33 piG′1 ∥∥∥∥∥∥ ≤ ∞∑ j=1 ∥∥∥∥ ∂r+p∂kr1∂kp2 piG′1∆−1k X(j)33 ∥∥∥∥ ≤ Cε,r,p|z|R2 ∞∑ j=1 (j + 2)r+p ( 4 9 )j+1 1 2 (j2 + 3j) ≤ CCε,r,p 1|z|R2 , where C is an universal constant. Therefore, ∣∣∣∣ ∂r+p∂kr1∂kp2R3(k) ∣∣∣∣ = ∣∣∣∣∣∣F{d′}G′1 ∂ r+p ∂kr1∂k p 2 ∆−1k ∞∑ j=1 X (j) 33 GG′1{d′} ∣∣∣∣∣∣ ≤ ‖F{d′}G′1‖ ∥∥∥∥∥∥piG′1 ∂ r+p ∂kr1∂k p 2 ∆−1k ∞∑ j=1 X (j) 33 piG′1 ∥∥∥∥∥∥ ‖GG′1{d′}‖ ≤ CCε,r,p‖f‖l1‖g‖l1 1 |z|R2 . Finally, combining all the estimates we have∥∥∥∥ ∂n+m∂kn1 ∂km2 α(3)µ,d′(k) ∥∥∥∥ ≤ 4∑ j=1 ∥∥∥∥ ∂n+m∂kn1 ∂km2 Rj(k) ∥∥∥∥ ≤ 3 C|z|R2 + C |z|3R ≤ 4C|z|R2 , where C = Cε,Λ,A,q,f,g,m,n is a constant. Set ρε,A,q,m,n := max{ρ1;ε,m,n, ρ2;ε,A,q,m,n}. The proof of the lemma for j = 3 is complete. 112 4.6 The regular piece Proof of Theorem 3.4.1. Step 1 (defining equation) We first derive a defining equation for the Fermi curve. Without loss of generality we can assume that Â(0) = 0 (see the discussion in §3.1). LetG = {0}, recall thatG′ = Γ#\{0}, and consider the region (Tν(0)\∪b∈G′Tb)\Kρ, where ρ is a constant to be chosen sufficiently large obeying ρ ≥ R. By Proposition 4.2.1(i) we have G′ = {b ∈ Γ# | |Nb(k)| ≥ ε|v|}. To simplify the notation write Mν := ( F̂(A, V ) ∩ Tν(0) ) \ Kρ ∪ ⋃ b∈Γ#\{0} Tb . By Lemma 4.2.4(i), a point k is in Mν if and only if N0(k) +D0,0(k) = 0. By Proposition 4.3.1, if we set w(k) := wν,0(k) = k1 + i(−1)νk2 and z(k) := zν,0(k) = k1 − i(−1)νk2, this equation becomes β1w 2 + β2z2 + (1 + β3)wz + β4w + β5z + β6 + q̂(0) = 0, (4.6.1) where β1 := J00ν′ , β2 := J 00 ν , β3 := K 00, β4 := L00ν′ , β5 := L 00 ν , β6 := M 00 − q̂(0), with J00ν , K 00, L00ν and M 00 given by Proposition 4.3.1. Observe that all the coefficients β1, . . . , β6 have exactly the same form as the function Φ0,0(k) of Lemma 4.4.1(i) (see (4.4.1)). Thus, by this lemma, for 1 ≤ i ≤ 6 we have βi = β (1) i + β (2) i + β (3) i , (4.6.2) where the functions β(j)i are analytic in the region under consideration with |β(j)i (k)| ≤ C (2|z(k)| − ρ)j ≤ C |z(k)|j for 1 ≤ j ≤ 2 and |β (3) i (k)| ≤ C |z(k)|ρ2 , 113 where C = Cε,Λ,q,A is a constant. The exact expressions for β (j) i can be easily obtained from the definitions and from Lemma 4.4.1(i). Substituting (4.6.2) into (4.6.1) and dividing both sides of the equation by z yields w + β(1)2 z + g = 0, (4.6.3) where g := β1w 2 z + (β(2)2 + β (3) 2 )z + β3w + β4w z + β5 + β6 z + q̂(0) z (4.6.4) obeys |g(k)| ≤ C ρ (4.6.5) with a constant C = Cε,Λ,q,A. Therefore, a point k is in Mν if and only if F (k) = 0, where F (k) := w(k) + β(1)2 (k) z(k) + g(k) is an analytic function (in the region under consideration). Step 2 (candidates for a solution) Let us now identify which points are candidates to solve the equation F (k) = 0. First observe that, by Proposition 3.2.2(c) we have N1(0) ∩N2(d) = {(iθ1(d), θ1(d))} and N1(d) ∩N2(0) = {(iθ2(d),−θ2(d))}. Thus, the lines Nν(0) and Nν′(d) intersect at Nν(0) ∩Nν′(d) = {(iθν(d), (−1)ν′θν(d))}. Hence, the second coordinate of this point and the second coordinate of a point k differ by pr(k)− pr(Nν(0) ∩Nν′(d)) = k2 − (−1)ν′θν(d) = k2 + (−1)νθν(d). Now observe that, if k ∈ Tν(0) ∩ Tν′(d) then |k1 + i(−1)νk2| < ε 114 and |k2 + (−1)νθν(d)| = |i(−1)νk2 + iθν(d)| = ∣∣i(−1)νk2 + 12(i(−1)νd2 − d1)∣∣ = ∣∣1 2(k1 + i(−1)νk2)− 12(k1 + d1 − i(−1)ν(k2 + d2) ∣∣ ≤ 12 ∣∣N0,ν(k)−Nd,ν′(k)∣∣ < ε2 + ε2 = ε. That is, the second coordinate of k and the second coordinate of Nν(0) ∩ Nν′(d) must be apart from each other by at most ε. This gives a necessary condition on the second coordinate of a point k for being in Mν . Conversely, if a point k is in the (ε/4)-tube inside Tν(0), that is, |k1 + i(−1)νk2| < ε4 , and its second coordinate differ from the second coordinate of Nν(0) ∩ Nν′(d) by at most ε/4, that is, |k2 + (−1)νθν(d)| < ε4 , then |Nd,ν′(k)| = ∣∣N0,ν(k)− 2(k2 + (−1)νθν(d))| ≤ ε4 + 2 ε4 < ε, that is, the point k is also in Tν′(d) and hence lie in the intersection Tν(0) ∩ Tν′(d). This gives a sufficient condition on the first and second coordinates of a point k for being in Tν(0) ∩ Tν′(d). For y ∈ C define the set of candidates for a solution of F (k) = 0 as Mν(y) := pr−1(y) ∩ Tν(0) \ ⋃ b∈Γ#\{0} Tb = pr−1(y) ∩ Tν(0) \ ⋃ b∈Γ#\{0} Tν′(b) . Observe that, if |y + (−1)νθν(b)| ≥ ε for all b ∈ Γ# \ {0} then Mν(y) = pr−1(y) ∩ Tν(0) = {(k1, y) ∈ C2 | |k1 + i(−1)νy| < ε}. (4.6.6) On the other hand, if |y+ (−1)νθν(d)| < ε for some d ∈ Γ# \ {0}, then there is at most one such d and consequently Mν(y) = pr−1(y) ∩ (Tν(0) \ Tν′(d)) = {(k1, y) ∈ C2 | |k1 + i(−1)νy| < ε and |k1 + d1 + i(−1)ν′(y + d2)| ≥ ε}. (4.6.7) 115 Indeed, suppose there is another d′ 6= 0 such that |y + (−1)νθν(d′)| < ε. Then, |d− d′| = |2(−1)νθν(d− d′)| = |y + (−1)νθν(d)− (y + (−1)νθν(d′))| ≤ 2ε < 2Λ, which contradicts the definition of Λ. Thus, there is no such d′ 6= 0. Step 3 (uniqueness) We now prove that, given k2, if there exists a solution k1(k2) of F (k1, k2) = 0, then this solution is unique and it depends analytically on k2. This follows easily using the implicit function theorem and the estimates below, which we prove later. Proposition 4.6.1. Under the hypotheses of Theorem 3.4.1 we have |F (k)− w(k)| ≤ ε 900 + C1 ρ (a) and ∣∣∣∣ ∂F∂k1 (k)− 1 ∣∣∣∣ ≤ 17 · 34 + C2ρ , (b) where the constants C1 and C2 depend only on ε, Λ, q and A. Now, suppose that (k1, y) ∈Mν(y). Then,∣∣∣∣ ∂F∂k1 (k1, y)− 1 ∣∣∣∣ ≤ 17 · 34 + C2ρ . Hence, by the implicit function theorem [16, Theorem 3.7.1], by choosing the constant ρ ≥ R sufficiently large, if F (k∗1, y) = 0 for some (k∗1, y) ∈ Mν(y), then there is a neighbourhood U × V ⊂ C2 which contains (k∗1, y), and an analytic function η : V → U such that F (k1, k2) = 0 for all (k1, k2) ∈ U × V if and only if k1 = η(k2). In particular, this implies that the equation F (k1, k2) = 0 has at most one solution (η(y), y) in Mν(y) for each y ∈ C. We next look for conditions on y to have a solution or have no solution in Mν(y). Step 4 (existence) We first state an improved version of Proposition 4.6.1(a). Proposition 4.6.2. Under the hypotheses of Theorem 3.4.1 we have F (k)− w(k) = β(1,0)2 + β(1,1)2 (w(k)) + β(1,2)2 (k) + h(k), where β (1,0) 2 = −2i ∑ b,c∈G′1 θν′(Â(−b)) θν′(b) [ δb,c + θν′(Â(b− c)) θν′(c) ] θν(Â(c)) (4.6.8) is a constant that depends only on ρ and A and h := β(1,3)2 + g. 116 Furthermore, |β(1,0)2 | < 1 100Λ ε2, |β(1,1)2 (k)| < 1 40Λ2 ε3, |β(1,2)2 (k)| < 1 74Λ3 ε4, |h(k)| ≤ Cε,Λ,q,A 1 ρ . We now derive conditions for the existence of solutions. Suppose that F (η(y), y) = 0. Then, since η(y)+ i(−1)νy = w(η(y), y) and ε < Λ/6, using the above proposition we obtain |η(y) + i(−1)νy| = |w(η(y), y)| = |F (η(y), y)− w(η(y), y)| ≤ ε 2 100Λ + ε3 40Λ2 + ε4 74Λ3 + C ρ ≤ ε 2 50Λ + C ρ . Hence, by choosing the constant ρ sufficiently large we find that |η(y) + i(−1)νy| < ε 2 40Λ . In view of (4.6.7), there is no solution in Mν(y) if for some d ∈ Γ# \ {0} we have |y + (−1)νθν(d)| < ε and |η(y) + d1 + i(−1)ν′(y + d2)| < ε. This happens if |y + (−1)νθν(d)| ≤ 12 ( ε− ε 2 40Λ ) because in this case |η(y) + d1 + i(−1)ν′(y + d2)| = |η(y) + i(−1)νy − 2i(−1)νy + d1 − i(−1)νd2| ≤ |η(y) + i(−1)νy|+ 2|y + (−1)νθν(d)| < ε. Therefore, the image set of pr is contained in Ω1 := { z ∈ C ∣∣∣∣∣ |z + (−1)νθν(b)| > 12 ( ε− ε 2 40Λ ) for all b ∈ Γ# \ {0} } . On the other hand, in view of (4.6.6), there is a solution in Mν(y) if |y + (−1)νθν(b)| > ε for all b ∈ Γ# \ {0}. Recall from Proposition 4.4.4(a) that ρ < |v| < 8|k2|. Thus, the image set of pr contains the set Ω2 := { z ∈ C ∣∣∣ 8|z| > ρ and |z + (−1)νθν(b)| > ε for all b ∈ Γ# \ {0}}. Step 5 (conclusion) Summarizing, we have the following biholomorphic correspondence: Mν 3 k pr−−−−−−−−→ k2 ∈ Ω, Mν 3 (η(y), y) pr −1 ←−−−−−−−−−− y ∈ Ω, 117 where Ω2 ⊂ Ω ⊂ Ω1 and η(y) = −β(1,0)2 − i(−1)νy − r(y), with the constant β(1,0)2 given by (4.6.8), |β(1,0)2 | < ε2 100Λ and |r(y)| ≤ ε 3 50Λ2 + C ρ . This completes the proof of the theorem. Proof of Propositions 4.6.1 and 4.6.2 We follow the same notation as above. Proof of Proposition 4.6.1. (a) Recall that β2 = J00ν . First observe that, by Proposition 4.3.1, Lemma 4.4.1, and (4.4.25), we have β (1) 2 (k) = (J 00 ν ) (1)(k) = ∑ b,c∈G′1 (1, i(−1)ν) · Â(−b) Nb(k) Sb,c (1,−i(−1)ν) · Â(c). (4.6.9) Thus, by (4.5.11) and (4.5.16), |β(1)2 (k)| ≤ √ 2‖Â‖l1 2 Λ(2|z(k)| −R) 45 44 √ 2‖Â‖l1 ≤ 4 Λ|z(k)| 44 45 ( 2ε 63 )2 ≤ ε 900 1 |z(k)| . (4.6.10) Now recall that |g(k)| ≤ Cε,Λ,q,A 1 ρ . Hence, |F (k)− w(k)| = |β(1)2 (k)z(k) + g(k)| ≤ ε 900 + Cε,Λ,q,A 1 ρ . This proves part (a). (b) We first compute ∂g ∂k1 = ∂ ∂k1 ( β1w 2 z + (β(2)2 + β (3) 3 )z + β3w + β4w z + β5 + β6 z + q̂(0) z ) = ∂β1 ∂k1 w2 z + β1 2wz − w2 z2 + ( ∂β (2) 2 ∂k1 + ∂β (3) 2 ∂k1 ) z + β(2)2 + β (3) 2 + ∂β3 ∂k1 w + β3 + ∂β4 ∂k1 w z + β4 z − w z2 + ∂β5 ∂k1 + ∂β6 ∂k1 1 z − β6 z2 − q̂(0) z2 . (4.6.11) 118 Now observe that, since k ∈ Tν(0) \ Kρ we have |w(k)| < ε, 3|v| ≥ |z| and ρ < |v| ≤ |z|. Furthermore, by Lemmas 4.4.1(i), 4.5.1(i) and 4.5.3(i), for 1 ≤ i ≤ 6 and 1 ≤ j ≤ 2, |βi(k)| ≤ C|z(k)| , |β (j) i (k)| ≤ C |z(k)|j , |β (3) i (k)| ≤ C |z(k)|ρ2 ,∣∣∣∣∂βi(k)∂k1 ∣∣∣∣ ≤ C|z(k)| , ∣∣∣∣∣∂β(j)i (k)∂k1 ∣∣∣∣∣ ≤ C|z(k)|j , ∣∣∣∣∣∂β(3)i (k)∂k1 ∣∣∣∣∣ ≤ C|z(k)|ρ2 , (4.6.12) where C = Cε,Λ,q,A in all cases. Hence,∣∣∣∣∂g(k)∂k1 ∣∣∣∣ ≤ Cε,Λ,q,A 1ρ. (4.6.13) By Lemma 4.5.3(i) with f = g = (1,−i(−1)ν) · Â, we obtain∣∣∣∣∣z(k)∂β(1)2 (k)∂k1 ∣∣∣∣∣ ≤ |z(k)| 13Λ2|z(k)|‖(1,−i(−1)ν) · Â‖2l1 ≤ 26 Λ2 ‖Â‖2l1 ≤ 26 Λ2 4ε2 (63)2 ≤ 26 · 4 (63)2 1 62 < 1 7 · 34 . (4.6.14) Therefore,∣∣∣∣ ∂F∂k1 (k)− 1 ∣∣∣∣ = ∣∣∣∣ ∂∂k1 (F (k)− w(k)) ∣∣∣∣ = ∣∣∣∣ ∂∂k1 (β(1)2 (k)z(k) + g(k)) ∣∣∣∣ = ∣∣∣∣∣∂β(1)2∂k1 (k)z(k) + β(1)2 (k) + ∂g∂k1 (k) ∣∣∣∣∣ ≤ 17 · 34 + Cε,Λ,q,A 1ρ. This proves part (b) and completes the proof of the proposition. Here is the proof of Proposition 4.6.2. Proof of Proposition 4.6.2. First observe that (1, i(−1)ν) ·A = A1 + i(−1)νA2 = A1 − i(−1)ν′A2 = −2i 12(iA1 + (−1) ν′A2) = −2iθν′(A). Thus, recalling (4.6.9), β (1) 2 (k) = (J 00 ν ) (1)(k) = ∑ b,c∈G′1 2iθν′(Â(−b)) Nb(k) Sb,c 2iθν(Â(c)). Now, by Lemma 4.4.2 we have z(k)β(1)2 (k) = β (1,0) 2 + β (1,1) 2 (w(k)) + β (1,2) 2 (k) + β (1,3) 3 (k), where β (1,0) 2 = −2i ∑ b,c∈G′1 θν′(Â(−b)) θν′(b) [ δb,c + θν′(Â(b− c)) θν′(c) ] θν(Â(c)) 119 and |β(1,3)3 (k)| ≤ CΛ,A 1 |z(k)| < CΛ,A 1 ρ . Hence, F (k)− w(k) = z(k)β(1)2 (k) + g(k) = β(1,0)2 + β(1,1)2 (w(k)) + β(1,2)2 (k) + h(k) with h := β(1,3)3 + g. Furthermore, in view of (4.6.5), |h(k)| ≤ |β(1,3)3 (k)|+ |g(k)| < Cε,Λ,q,A 1 ρ . This proves the first part of the proposition. Finally, by (4.4.40), since ‖Â‖l1 < 2ε/63 and ε < Λ/6, we find that |β(1,0)2 | ≤ 1 2Λ ( 1 + 1 2Λ ‖θν′(Â)‖l1 ) ‖2iθν′(Â)‖l1‖2iθν(Â)‖l1 ≤ 1 2Λ ( 1 + 1 2Λ 2ε 63 ) 4‖Â‖2l1 ≤ 4 Λ ‖Â‖2l1 < 1 100Λ ε2 and |β(1,1)2 | ≤ ε Λ2 ( 1 + 7 6Λ ‖θν′(Â)‖l1 ) ‖2iθν′(Â)‖l1‖2iθν(Â)‖l1 ≤ ε Λ2 ( 1 + 7 6Λ 2ε 63 ) 4‖Â‖2l1 ≤ 8 Λ2 ε‖Â‖2l1 < 1 40Λ2 ε3 and |β(1,2)2 | ≤ 64 Λ3 ‖θν′(Â)‖2l1‖2iθν′(Â)‖l1‖2iθν(Â)‖l1 ≤ 256 Λ3 ‖Â‖4l1 < 1 74Λ3 ε4. This completes the proof. 4.7 The handles Proof of Theorem 3.4.2. Step 1 (defining equation) We first derive a defining equation for the Fermi curve. Without loss of generality we can assume that Â(0) = 0. Let G = {0, d}, recall that G′ = Γ# \ {0, d}, and consider the region (Tν(0) ∩ Tν′(d)) \ Kρ, where ρ is a constant to be chosen sufficiently large obeying ρ ≥ R. Observe that, this requires d being sufficiently large for (Tν(0)∩ Tν′(d)) \Kρ being not empty. In fact, by Proposition 4.4.4(ii), for k in this region we have ρ < |v| ≤ 2|d|. 120 Now, recall from Proposition 4.2.1(ii) that G′ = {b ∈ Γ# | |Nb(k)| ≥ ε|v|}, and to simplify the notation write Hν := F̂(A, V ) ∩ (Tν(0) ∩ Tν′(d)) \ Kρ. By Lemma 4.2.4(ii), a point k is in Hν if and only if (N0(k) +D0,0(k))(Nd(k) +Dd,d(k))−D0,d(k)Dd,0(k) = 0. (4.7.1) Define w1(k) := wν,0 = k1 + i(−1)νk2, z1(k) := zν,0 = k1 − i(−1)νk2, w2(k) := wν′,d = k1 + d1 + i(−1)ν′(k2 + d2), z2(k) := zν′,d = k1 + d1 − i(−1)ν′(k2 + d2). (4.7.2) Note that, by Proposition 4.4.4(ii), |v| ≤ |z1| ≤ 3|v|, |v| ≤ |z2| ≤ 3|v| and |d| ≤ |z2| ≤ 2|d|. By Proposition 4.3.1, N0 +D0,0 = β1w21 + β2z 2 1 + (1 + β3)w1z1 + β4w1 + β5z1 + β6 + q̂(0), Nd +Dd,d = η1w22 + η2z 2 2 + (1 + η3)w2z2 + η4w2 + η5z2 + η6 + q̂(0), (4.7.3) where β1 := J00ν′ , β2 := J 00 ν , β3 := K 00, β4 := L00ν′ , β5 := L 00 ν , β6 := M 00 − q̂(0), and η1 := Jddν , η2 := J dd ν′ , η3 := K dd, η4 := Lddν , η5 := L dd ν′ , η6 := M dd − q̂(0), with Jd ′d′ ν , K d′d′ , Ld ′d′ ν and M d′d′ given by Proposition 4.3.1. Observe that all the coefficients β1, . . . , β6 and η1, . . . , η6 have exactly the same form as the function Φd′,d′(k) of Lemma 4.4.1(ii) (see (4.4.1)). Thus, by this lemma, for 1 ≤ i ≤ 6 we have βi = β (1) i + β (2) i + β (3) i , ηi = η (1) i + η (2) i + η (3) i , (4.7.4) 121 where the functions β(j)i and η (j) i are analytic in the region under consideration with |β(j)i (k)| ≤ C (2|z1(k)| − ρ)j ≤ C |z1(k)|j for 1 ≤ j ≤ 2 and |β (3) i (k)| ≤ C |z1(k)|ρ2 , |η(j)i (k)| ≤ C (2|z2(k)| − ρ)j ≤ C |z2(k)|j for 1 ≤ j ≤ 2 and |η (3) i (k)| ≤ C |z2(k)|ρ2 , where C = Cε,Λ,q,A is a constant. The exact expressions for β (j) i and η (j) i can be easily obtained from the definitions and from Lemma 4.4.1(ii). Substituting (4.7.4) into (4.7.3) yields 1 z1 (N0 +D0,0) = w1 + β (1) 2 z1 + g1, 1 z2 (Nd +Dd,d) = w2 + η (1) 2 z2 + g2, (4.7.5) where g1 := β1w 2 1 z1 + (β(2)2 + β (3) 2 )z1 + β3w1 + β4w1 z1 + β5 + β6 z1 + q̂(0) z1 , g2 := η1w 2 2 z2 + (η(2)2 + η (3) 2 )z2 + η3w2 + η4w2 z2 + η5 + η6 z2 + q̂(0) z2 (4.7.6) obey |g1(k)| ≤ C ρ and |g2(k)| ≤ C ρ (4.7.7) with a constant C = Cε,Λ,q,A. This gives us more information about the first term in (4.7.1). We next consider the second term in that equation. Write D0,d = c1(d) + p1 and Dd,0 = c2(d) + p2 (4.7.8) with c1(d) := q̂(−d)− 2d · Â(−d), p1 := D0,d − q̂(−d) + 2d · Â(−d), c2(d) := q̂(d) + 2d · Â(d), p2 := Dd,0 − q̂(d)− 2d · Â(d). We shall shortly prove the following estimates. Proposition 4.7.1. Under the hypotheses of Theorem 3.4.2 we have, for any integers n and m with n+m ≥ 0 and for 1 ≤ j ≤ 2,∣∣∣∣ ∂n+m∂kn1 ∂km2 pj(k) ∣∣∣∣ ≤ C1|d| and |cj(d)| ≤ C2|d| , where the constants C1 and C2 depend only on ε, Λ, q and A. 122 Thus, by dividing both sides of (4.7.1) by z1z2 and substituting (4.7.5) and (4.7.8) we find that 0 = 1 z1z2 [ (N0 +D0,0)(Nd +Dd,d)−D0,dDd,0 ] = (w1 + β (1) 2 z1 + g1)(w2 + η (1) 2 z2 + g2)− 1 z1z2 (c1(d) + p1)(c2(d) + p2). (4.7.9) We now introduce a (nonlinear) change of variables in C2. Set x1(k) := w1(k) + β (1) 2 (k) z1(k) + g1(k), x2(k) := w2(k) + η (1) 2 (k) z2(k) + g2(k). (4.7.10) We shall prove below that this transformation satisfies the following estimates. Proposition 4.7.2. Under the hypotheses of Theorem 3.4.2 we have: (i) For 1 ≤ j ≤ 2 and for ρ sufficiently large, |xj(k)− wj(k)| ≤ ε900 + C ρ < ε 8 . (ii) ∂x1∂k1 ∂x1∂k2 ∂x2 ∂k1 ∂x2 ∂k2 = 1 i(−1)ν 1 i(−1)ν′ (I +M) and ∂k1∂x1 ∂k1∂x2 ∂k2 ∂x1 ∂k2 ∂x2 = 1 2 1 1 i(−1)ν′ i(−1)ν (I +N) with ‖M‖ ≤ 4 7 · 34 + C ρ < 1 2 and ‖N‖ ≤ 4‖M‖. Furthermore, for all m, i, j ∈ {1, 2},∣∣∣∣ ∂2km∂xi∂xj ∣∣∣∣ ≤ 3Λ3 ε2 + Cρ . Here, all the constants C depend only on ε, Λ, q and A. By the inverse function theorem, these estimates imply that the above transformation is invertible. Therefore, by rewriting the equation (4.7.9) in terms of these new variables, we conclude that a point k is in Hν if and only if x1(k) and x2(k) satisfy the equation x1x2 + r(x1, x2) = 0, (4.7.11) 123 where r(x1, x2) := − 1 z1z2 (c1(d) + p1)(c2(d) + p2). This is the defining equation we shall study in detail below. In order to do this we need some estimates. Step 2 (estimates) Using the above inequalities we have, for i, j, l ∈ {1, 2}, ∣∣∣∣ ∂∂xi pj(k(x)) ∣∣∣∣ ≤ 2∑ m=1 ∣∣∣∣ ∂pj∂km ∂km∂xi ∣∣∣∣ ≤ C|d| and ∣∣∣∣ ∂2∂xi∂xl pj(k(x)) ∣∣∣∣ ≤ 2∑ m,n=1 ∣∣∣∣ ∂2pj∂km∂kn ∂km∂xi ∂kn∂xl ∣∣∣∣+ 2∑ m=1 ∣∣∣∣ ∂pj∂km ∂ 2km ∂xi∂xl ∣∣∣∣ ≤ C|d| , so that |r(x)| ≤ C 1|d|2 1 |d| 1 |d| ≤ C |d|4 ,∣∣∣∣ ∂∂xi r(x) ∣∣∣∣ ≤ C 1|d|3 1|d| 1|d| + C 1|d|2 1|d| 1|d| ≤ C|d|4 and ∣∣∣∣ ∂2∂xi∂xj r(x) ∣∣∣∣ ≤ C|d|4 . Here, all the constants depend only on ε, Λ, q and A. Step 3 (Morse lemma) We now apply the quantitative Morse lemma in §4.8 for studying the equation (4.7.11). We consider this lemma with a = b = C |d|4 , δ = ε, and d sufficiently large so that b < max{23 155 , ε4}. Observe that, under this condition we have (δ − a)(1− 19b) > ε 2 and (δ − a)(1− 55b) > ε 4 . According to this lemma, there is a biholomorphism Φν defined on Ω1 := { (z1, z2) ∈ C2 ∣∣∣ |z1| < ε2 and |z2| < ε2} with range containing { (x1, x2) ∈ C2 ∣∣∣ |x1| < ε4 and |x2| < ε4} (4.7.12) 124 such that ‖DΦν − I‖ ≤ C|d|2 , ((x1x2 + r) ◦ Φν)(z1, z2) = z1z2 + td, |td| ≤ C|d|4 , |Φν(0)| ≤ C|d|4 , (4.7.13) where DΦν is the derivative of Φν and td is a constant that depends on d. Hence, if for ν = 1 we define φd,1 : Ω1 −→ T1(0) ∩ T2(d) as φd,1(z1, z2) := (k1(Φ1(z1, z2)), k2(Φ1(z1, z2))), where k(x) is the inverse of the transformation (4.7.10), we obtain the desired map. Note that the conclusion (ii) of the theorem is immediate. We next prove (i) and (iii). Step 4 (proof of (i)) By Proposition 4.7.2(i), for 1 ≤ j ≤ 2, |xj(k)− wj(k)| ≤ ε8 . Now, recall from (4.7.2) the definition of w1(k) and w2(k). Then, since |xj(k)| ≤ |xj(k)− wj(k)|+ |wj(k)| < ε8 + |wj(k)|, the set { (k1, k2) ∈ C2 ∣∣∣ |w1(k)| < ε8 and |w2(k)| < ε8} is contained in the set (4.7.12). This proves the first part of (i). To prove the second part we use Proposition 4.7.2 and (4.7.13). First observe that Dφd,1 = ∂k ∂x DΦ1 = 1 2 1 1 i −i (I +N)(I +DΦ1 − I) = 1 2 1 1 i −i (I +N +R), where ‖N‖ ≤ 1 33 + C ρ and ‖R‖ ≤ C|d|2 . 125 Furthermore, from (4.7.2) and (4.7.10) we have k1 = iθν(d) + 1 2 (w1 + w2) = iθν(d) + 1 2 (x1 + x2 + β (1) 2 z1 + η (1) 2 z2 + g1 + g2), k2 = −(−1)νθν(d) + (−1) ν 2i (w1 − w2) = −(−1)νθν(d) + (−1) ν 2i (x1 − x2 − β(1)2 z1 + η(1)2 z2 − g1 + g2), so that φd,1(0) = k(Φ1(0)) = k ( O ( 1 |d|4 )) = (iθν(d),−(−1)νθν(d)) +O ( ε 900 ) +O ( 1 ρ ) . Step 5 (proof of (iii)) To prove part (iii) it suffices to note that T1(0)∩T2(d)∩F̂(A, V ) is mapped to T1(−d) ∩ T2(0) ∩ F̂(A, V ) by translation by d and define φd,2 by φd,2(z1, z2) := φd,1(z2, z1) + d. This completes the proof of the theorem. Proof of Propositions 4.7.1 and 4.7.2 We follow the same notation as above. Proof of Proposition 4.7.1. It suffices to estimate cd′,d′′ := q̂(d′ − d′′)− 2(d′ − d′′) · Â(d′ − d′′) and pd′,d′′ := Dd′,d′′ − cd′,d′′ for d′, d′′ ∈ {0, d} with d′ 6= d′′. Define ld ′d′′ ν := (1, i(−1)ν) · Â(d′ − d′′). Observe that, since |q̂(d′ − d′′)| = 1|d′ − d′′|2 |d ′ − d′′|2 |q̂(d′ − d′′)| ≤ 1|d′ − d′′|2 ∑ b∈Γ# |b|2 |q̂(b)| ≤ ‖b2q̂(b)‖l1 1 |d|2 , and similarly |Â(d′ − d′′)| ≤ ‖b2Â(b)‖l1 1 |d|2 , it follows that |cd′,d′′ | ≤ CA,q|d| and |l d′d′′ ν | ≤ CA |d|2 . This gives the desired bounds for c1 and c2. 126 Now, by Proposition 4.3.1 we have p = Jd ′d′′ ν′ w 2 ν,d′ +J d′d′′ ν z 2 ν,d′ +K d′d′′wν,dzν,d′ + (L̃d ′d′′ ν′ − ld ′d′′ ν′ )wν,d′ + (L̃ d′d′′ ν − ld ′d′′ ν )zν,d′ + M̃ d′d′′ with L̃d ′d′′ ν := L d′d′′ ν + l d′d′′ ν and M̃ d′d′′ := Md ′d′′ − c. Observe that all the coefficients Jd ′d′′ ν , K d′d′′ , L̃d ′d′′ ν and M̃ d′d′′ have exactly the same form as the function Φd′,d′′(k) of Lemma 4.5.2 (see Proposition 4.3.1 and (4.4.1)). Thus, by this lemma with β = 2, for any integers n and m with n + m ≥ 0, the absolute value of the ∂n+m ∂kn1 ∂k m 2 -derivative of each of these functions is bounded above by Cε,Λ,A,q,m,n 1 |d|3 . Hence, if we recall from Proposition 4.4.4(ii) that |z1(k)| ≤ 6|d| and |z2(k)| ≤ 2|d|, and apply the Leibniz rule we find that∣∣∣∣ ∂n+m∂kn1 ∂km2 pd′,d′′(k) ∣∣∣∣ ≤ Cm,n C|d| . This yields the desired bounds for p1 and p2 and completes the proof. We now prove Proposition 4.7.2. Proof of Proposition 4.7.2. (i) Similarly as in (4.6.10) we have |β(1)2 (k)| ≤ ε 900 1 |z1(k)| and |η (1) 2 (k)| ≤ ε 900 1 |z2(k)| . Thus, in view of (4.7.7), and by choosing ρ sufficiently large, |x1(k)− w1(k)| ≤ |β(1)2 (k) z1(k) + g1(k)| ≤ ε 900 + C ρ < ε 8 , |x2(k)− w2(k)| ≤ |η(1)2 (k) z2(k) + g2(k)| ≤ ε 900 + C ρ < ε 8 . This proves part (i). (ii) Recall (4.7.2) and (4.7.10). Then, for 1 ≤ j ≤ 2, ∂x1 ∂kj = ∂ ∂kj (w1 + z1β (1) 2 + g1) = ∂w1 ∂kj + z1 ∂β (1) 2 ∂kj + ∂z1 ∂kj β (1) 2 + ∂g1 ∂kj , ∂x2 ∂kj = ∂ ∂kj (w2 + z2η (1) 2 + g2) = ∂w2 ∂kj + z2 ∂η (1) 2 ∂kj + ∂z2 ∂kj η (1) 2 + ∂g2 ∂kj . 127 To estimate the terms above on the right hand side we use a calculation that we have already done in the proof of Proposition 4.6.1. First observe that the functions g1 and g2 are similar to the function g (see (4.7.6) and (4.6.4)). Thus, it is easy to see that ∂g1∂kj and ∂g2 ∂kj are given by expressions similar to (4.6.11). Since k ∈ Tν(0)∩Tν′(d) we have |w1(k)| < ε and |w2(k)| < ε. Recall also the inequalities in Proposition 4.4.4(ii). Hence, by Lemmas 4.4.1(ii), 4.5.1(ii) and 4.5.3(ii), we obtain (4.6.12) with k1 and z(k) replaced by kj and z1(k), respectively, and for k1, z(k) and β replaced by kj , z2(k) and η, respectively. Consequently, similarly as in (4.6.13) and using again Lemma 4.4.1(ii), for 1 ≤ j ≤ 2 we have∣∣∣∣∂z1∂kj β(1)2 + ∂g1∂kj ∣∣∣∣ ≤ Cε,Λ,q,A 1ρ and ∣∣∣∣∂z2∂kj η(1)2 + ∂g2∂kj ∣∣∣∣ ≤ Cε,Λ,q,A 1ρ. Now recall that β2 = J00ν and η2 = J dd ν′ . Then, by Proposition 4.3.1, Lemma 4.4.1(ii), and (4.4.25), it follows that β (1) 2 (k) = (J 00 ν ) (1)(k) = ∑ b,c∈G′1 (1, i(−1)ν) · Â(−b) Nb(k) Sb,c (1,−i(−1)ν) · Â(c), η (1) 2 (k) = (J dd ν′ ) (1)(k) = ∑ b,c∈G′1 (1, i(−1)ν′) · Â(d− b) Nb(k) Sb,c (1,−i(−1)ν′) · Â(c− d). Hence, by Lemma 4.5.3(ii), similarly as in (4.6.14), for 1 ≤ j ≤ 2,∣∣∣∣∣z1(k)∂β(1)2 (k)∂kj ∣∣∣∣∣ ≤ 13Λ2 ‖(1,−i(−1)ν) · Â‖2l1 < 17 · 34 ,∣∣∣∣∣z2(k)∂η(1)2 (k)∂kj ∣∣∣∣∣ ≤ 13Λ2 ‖(1,−i(−1)ν′) · Â‖2l1 < 17 · 34 . Therefore, ∂x1∂k1 ∂x1∂k2 ∂x2 ∂k1 ∂x2 ∂k2 = 1 i(−1)ν 1 i(−1)ν′ + z1(k)∂β(1)2 (k)∂k1 z1(k)∂β(1)2 (k)∂k2 z2(k) ∂η (1) 2 (k) ∂k1 z2(k) ∂η (1) 2 (k) ∂k2 + β(1)2 −i(−1)νβ(1)2 η (1) 2 −i(−1)ν ′ η (1) 2 + ∂g1∂k1 ∂g1∂k2 ∂g2 ∂k1 ∂g2 ∂k2 =: 1 i(−1)ν 1 i(−1)ν′ (I +M1 +M2 +M3), where ‖M1‖ ≤ 2 27 · 34 and ‖M2 +M3‖ ≤ Cε,Λ,q,A 1 ρ . Set M := M1 +M2 +M3. This proves the first claim. 128 Now, by choosing ρ sufficiently large we can make ‖M‖ < 1 2 . Write P := 1 i(−1)ν 1 i(−1)ν′ . Then, by the inverse function theorem and using the Neumann series, ∂k1∂x1 ∂k1∂x2 ∂k2 ∂x1 ∂k2 ∂x2 = ∂x1∂k1 ∂x1∂k2 ∂x2 ∂k1 ∂x2 ∂k2 −1 = (I +M)−1P−1 = (I + M̃)P−1 =: P−1(I + PM̃P−1) = 1 2 1 1 i(−1)ν′ i(−1)ν (I + PM̃P−1), with ‖PM̃P−1‖ ≤ 2‖M̃‖1 ≤ 2‖M‖ 1− ‖M‖ ≤ 4‖M‖. Set N := PM̃P−1. This proves the second claim. Differentiating the matrix identity TT−1 = I and applying the chain rule we find that ∂2km ∂xi∂xj = − 2∑ l,p=1 ∂km ∂xl ∂ ∂xi ( ∂xl ∂kp ) ∂kp ∂xj = − 2∑ l,p=1 ∂km ∂xl ∂2xl ∂kr∂xp ∂kr ∂xi ∂kp ∂xj . Furthermore, in view of the above calculations we have∣∣∣∣ ∂ki∂xj ∣∣∣∣ ≤ 12(1 + ‖N‖) ≤ 12(1 + 4‖M‖) ≤ 1 2 ( 1 + 4 1 2 ) < 3 2 . Thus, ∣∣∣∣ ∂2km∂xi∂xj ∣∣∣∣ ≤ 4(32 )3 sup l,r,p ∣∣∣∣ ∂2xl∂kr∂xp ∣∣∣∣ . We now estimate ∂2x1 ∂ki∂kj = ∂z1 ∂ki ∂β (1) 2 ∂kj + z1 ∂2β (1) 2 ∂ki∂kj + ∂z1 ∂kj ∂β (1) 2 ∂ki + ∂2g1 ∂ki∂kj and ∂2x2 ∂ki∂kj = ∂z2 ∂ki ∂η (1) 2 ∂kj + z2 ∂2η (1) 2 ∂ki∂kj + ∂z2 ∂kj ∂η (1) 2 ∂ki + ∂2g2 ∂ki∂kj . 129 From (4.6.11) with g, w and z replaced by g1, w1 and z1, respectively, we obtain ∂2g1 ∂k21 = ∂2β1 ∂k21 w21 z1 + 2 ∂β1 ∂k1 2w1z1 − w21 z21 + β1 2z21 − 6w1z1 + 4w21 z31 + ( ∂2β (2) 2 ∂k21 + ∂2β (3) 2 ∂k21 ) z1 + 2 ( ∂β (2) 2 ∂k1 + ∂β (3) 2 ∂k1 ) + ∂2β3 ∂k21 w1 + 2 ∂β3 ∂k1 + ∂2β4 ∂k21 w1 z1 + 2 ∂β4 ∂k1 z1 − w1 z21 + β4 2(w1 − z1) z31 + ∂2β5 ∂k21 + ∂2β6 ∂k21 1 z1 − 2∂β6 ∂k1 1 z21 + 2 β6 z31 + 2q̂(0) z31 . Hence, by Lemmas 4.4.1(ii), 4.5.1(ii) and 4.5.3(ii),∣∣∣∣∂2g1∂k21 ∣∣∣∣ ≤ Cε,Λ,q,A 1ρ. Similarly we prove that ∣∣∣∣ ∂2gl∂ki∂kj ∣∣∣∣ ≤ Cε,Λ,q,A 1ρ for all l, i, j ∈ {1, 2} because all the derivatives acting on gl are essentially the same (up to constant factors). Indeed, ∂ ∂ki gl = ( ∂wl ∂ki ∂ ∂wl + ∂zl ∂ki ∂ ∂zl ) gl with ∂w1∂k1 ∂w1∂k2 ∂w2 ∂k1 ∂w2 ∂k2 = 1 i(−1)ν 1 i(−1)ν′ and ∂z1∂k1 ∂z1∂k2 ∂z2 ∂k1 ∂z2 ∂k2 = 1 −i(−1)ν 1 −i(−1)ν′ . In particular this implies ∣∣∣∣ ∂zi∂kj ∣∣∣∣ = 1. Furthermore, again by Lemma 4.5.3(ii),∣∣∣∣∣∂β(1)2∂kj ∣∣∣∣∣ ≤ Cε,Λ,q,A 1ρ, ∣∣∣∣∣∂η(1)2∂kj ∣∣∣∣∣ ≤ Cε,Λ,q,A 1ρ, and ∣∣∣∣∣z1(k)∂2β(1)2 (k)∂k1∂kj ∣∣∣∣∣ ≤ 65Λ3 ‖(1,−i(−1)ν) · Â‖2l1 < 15Λ3 ε2,∣∣∣∣∣z2(k)∂2η(1)2 (k)∂ki∂kj ∣∣∣∣∣ ≤ 65Λ3 ‖(1,−i(−1)ν′) · Â‖2l1 < 15Λ3 ε2. Hence, ∣∣∣∣ ∂2xl∂ki∂kj ∣∣∣∣ ≤ 15Λ3 ε2 + Cε,Λ,q,A 1ρ. Therefore, ∣∣∣∣ ∂2km∂xi∂xj ∣∣∣∣ ≤ 4(32 )3 sup l,r,p ∣∣∣∣ ∂2xl∂kr∂xp ∣∣∣∣ ≤ 3Λ3 ε2 + Cε,Λ,q,A 1ρ. This completes the proof of the proposition. 130 4.8 Quantitative Morse lemma In this section we prove a quantitative Morse lemma from [4] that is used above for proving Theorem 3.4.2. Lemma 4.8.1 (Quantitative Morse lemma [4]). Let δ be a constant with 0 < δ < 1 and assume that f(x1, x2) = x1x2 + r(x1, x2) is an holomorphic function on Dδ = {(x1, x2) ∈ C2 | |x1| ≤ δ and |x2| ≤ δ}. Suppose further that, for all x ∈ Dδ and 1 ≤ i ≤ 2 the function r satisfies∣∣∣∣ ∂r∂xi (x) ∣∣∣∣ ≤ a < δ and ∥∥∥∥∥ [ ∂2r ∂xi∂xj (x) ] i,j∈{1,2} ∥∥∥∥∥ ≤ b < 155 , where a and b are constants. Then f has a unique critical point ξ = (ξ1, ξ2) ∈ Dδ with |ξ1| ≤ a and |ξ2| ≤ a. Furthermore, let s = max{|ξ1|, |ξ2|}. Then there is a biholomorphic map Φ from the domain D(δ−s)(1−19b) to a neighbourhood of ξ ∈ Dδ that contains {(z1, z2) ∈ C2 | |zi − ξi| < (δ − s)(1− 55b) for 1 ≤ i ≤ 2} such that (f ◦ Φ)(z1, z2) = z1z2 + c, where c ∈ C is a constant fulfilling |c− r(0, 0)| ≤ a2. The differential DΦ obeys ‖DΦ− I‖ ≤ 18b. If ∂r∂x1 (0, 0) = 0 and ∂r ∂x2 (0, 0) = 0, then ξ = 0 and s = 0. Proof. Step 1 For 1 ≤ i ≤ 2 set Ci := { (x1, x2) ∈ Dδ ∣∣∣∣ ∂f∂xi (x) = 0 } . 131 To prove the first claim we show that C1 and C2 have a unique point of intersection. For each x1 with |x1| ≤ δ consider the functions u(x2) := x2 and v(x2) := ∂r ∂x1 (x1, x2) on the domain Ω := {x2 ∈ C | |x2| ≤ δ} with boundary ∂Ω := {x2 ∈ C | |x2| = δ}. By hypothesis, for all x2 ∈ ∂Ω we have |u(x2)| = δ and |v(x2)| ≤ a < δ, so that |u(x2)| > |v(x2)| and u(x2) 6= 0. Thus, by Rouche’s theorem [13, Theorem 10.43], the functions u(x2) and u(x2)+v(x2) have the same number of zeros in Ω. Since u(x2) has only one zero, the equation 0 = u(x2) + v(x2) = x2 + ∂r ∂x1 (x1, x2) = ∂f ∂x1 (x1, x2) has a unique solution x̃2(x1). By the implicit function theorem, this solution is an holomor- phic function of x1. Furthermore, differentiating with respect to x1 the identity x̃2(x1) = − ∂r ∂x1 (x1, x̃2(x1)) we find that ∂x̃2(x1) ∂x1 = −∂ 2r ∂x21 (x1, x̃2(x1)) ( 1 + ∂2r ∂x2∂x1 (x1, x̃2(x1)) )−1 . (4.8.1) Hence, using the hypotheses, |x̃2(x1)| ≤ a and ∣∣∣∣∂x̃2(x1)∂x1 ∣∣∣∣ ≤ b1− b . Similarly we can parametrize the curve C2 by a map x2 7→ (x̃1(x2), x2) satisfying |x̃1(x2)| ≤ a and ∣∣∣∣∂x̃1(x2)∂x2 ∣∣∣∣ ≤ b1− b . Therefore, the curves C1 and C2 intersect in a unique point (ξ1, ξ2) with x̃1(ξ2) = ξ1 and x̃2(ξ1) = ξ2. Thus we have |ξ1| ≤ a and |ξ2| ≤ a. This proves the first claim. Step 2 Without loss of generality we may assume that r(0, 0) = 0. (If r(0, 0) 6= 0 then instead of f consider f0(x1, x2) = x1x2+r0(x1, x2) with f0 := f−r(0, 0) and r0 := r−r(0, 0).) Define f̃(x1, x2) := f(x1 + ξ1, x2 + ξ2) 132 and r̃(x1, x2) := x1ξ2 + ξ1x2 + ξ1ξ2 + r(x1 + ξ1, x2 + ξ2). Then, f̃(x1, x2) = x1x2 + r̃(x1, x2), where r̃ is an holomorphic function on Dδ−s with s := max{|ξ1|, |ξ2|}. Since (ξ1, ξ2) is a critical point of f , we have r̃(0, 0) = ξ1ξ2 + r(ξ1, ξ2), ∂r̃ ∂x1 (0, 0) = ξ2 + ∂r ∂x1 (ξ1, ξ2) = 0, ∂r̃ ∂x2 (0, 0) = ξ1 + ∂r ∂x2 (ξ1, ξ2) = 0. Thus, if we translate the system of coordinates in C2 so that (ξ1, ξ2) is mapped to (0, 0), we obtain r̃(0, 0) = r(0, 0) = 0, ∂r̃ ∂x1 (0, 0) = 0 and ∂r̃ ∂x2 (0, 0) = 0. Furthermore, since ∂2r̃ ∂xi∂xj = ∂2r ∂xi∂xj , we still have the bound ∥∥∥∥∥ [ ∂2r̃ ∂xi∂xj (x) ] i,j∈{1,2} ∥∥∥∥∥ ≤ b. This shows that it suffices to prove the lemma in the special case that r(0, 0) = ∂r ∂x1 (0, 0) = ∂r ∂x2 (0, 0) = 0 and then replace δ by δ − s. Step 3 For 0 ≤ t ≤ 1 set ft(x1, x2) := x1x2 + t r(x1, x2). Below we construct a t-dependent vector field Xt(x) that is holomorphic on Dδ(1−4b) and satisfies r(x) + (∇ft)(x) ·Xt(x) = 0, (4.8.2) ‖Xt(x)‖ ≤ 5b(|x1|+ |x2|) ≤ 10bδ, (4.8.3)∥∥∥∥∂Xt∂xi (x) ∥∥∥∥ ≤ 8b for 1 ≤ i ≤ 2. (4.8.4) 133 Now, for 0 ≤ τ ≤ 1 consider the map Φτ : Dδ(1−10b) −→ C2, where Φτ (x) is the solution of the initial value problem d dτ Φτ (x) = Xτ (Φτ (x)) for 0 ≤ τ ≤ 1, Φ0(x) = x. This solution exists because Xτ (x) is holomorphic. Consequently, in view of (4.8.2), Φ0(x) = x and d dτ fτ (Φτ (x)) = r(Φτ (x)) + (∇fτ )(Φτ (x)) · d dτ Φτ (x) = 0. Furthermore, by (4.8.3) we have d dτ ‖Φτ (x)‖ = Φτ (x)‖Φτ (x)‖ · d dτ Φτ (x) = Φτ (x) ‖Φτ (x)‖ ·X τ (Φτ (x)) ≤ ‖Xτ (Φτ (x))‖ ≤ 10b‖Φτ (x)‖, so that ‖Φτ (x)‖ ≤ e10bτ‖Φ0(x)‖ = e10bτ‖x‖. This implies ∥∥∥∥ ddτ Φτ (x) ∥∥∥∥ = ‖Xτ (Φτ (x))‖ ≤ 10b‖Φτ (x)‖ ≤ 10be10b‖x‖. Hence, after integrating with respect to τ , ‖Φτ (x)− x‖ ≤ 10be10b‖x‖ ≤ 15bδ. This shows that Φt(x) remains in the domain of Xt(x) for all 0 ≤ t ≤ 1 and x ∈ Dδ(1−19b). Now observe that d dτ (DxΦτ )(x) = Dx(Xτ (Φτ (x))) = (DxXτ )(Φτ (x))(DxΦτ )(x), so that, by (4.8.4), d dτ ‖(DxΦτ )(x)‖ ≤ ∥∥∥∥ ddτ (DxΦτ )(x) ∥∥∥∥ ≤ ‖(DxXτ )(Φτ (x))‖ ‖(DxΦτ )(x)‖ ≤ √ 2 8b‖(DxΦτ )(x)‖. Consequently, ‖(DxΦτ )(x)‖ ≤ e12b‖(DxΦ0)(x)‖ = e12b,∥∥∥∥ ddτ (DxΦτ )(x) ∥∥∥∥ ≤ 18b 134 and ‖DxΦt − I‖ ≤ ∫ t 0 ∥∥∥∥ ddτ (DxΦτ )(x) ∥∥∥∥ dτ ≤ 18b < 1. Therefore, by the inverse function theorem, the map Φt is biholomorphic into its image. If we set Φ := Φ1, then Φ has the desired properties because f1 ◦ Φ1 = f1 ◦ Φ1 − f0 + f0 = ∫ 1 0 d dτ fτ (Φτ ) dτ + f0 = f0 and, by the contraction mapping theorem, the image of Dδ(1−19b) under Φ contains Bδ′ with δ′ = 1− 18b 1 + 18b (1− 19b)δ ≥ (1− 55b)δ. Step 4 To construct Xt(x) observe that the equation (4.8.2) is r(x1, x2) + 〈x2 + t ∂r∂x1 (x) x1 + t ∂r∂x2 (x) , Xt1(x) Xt2(x) 〉 = 0. (4.8.5) By the assumptions on r, for 1 ≤ i ≤ 2 we have∣∣∣∣ ∂r∂xi (x) ∣∣∣∣ ≤ b(|x1|+ |x2|) ≤ 2bδ. (4.8.6) Hence, |r(x)| ≤ b(|x1|+ |x2|)2 < 4bδ2 < 4bδ. Thus, since b < 1/55, by the inverse function theorem, for 0 ≤ t ≤ 1 the map Pt : Dδ −→ C2 (x1, x2) 7−→ ( x2 + t ∂r ∂x1 (x), x1 + t ∂r ∂x2 (x) ) is biholomorphic into its image, and the image contains Dδ(1−2b). Furthermore,∥∥∥∥∥∥DxPt − 0 1 1 0 ∥∥∥∥∥∥ = ∥∥∥∥t ∂2r∂xi∂xj ∥∥∥∥ ≤ tb (4.8.7) and ∥∥∥∥∥∥DxP−1t − 0 1 1 0 ∥∥∥∥∥∥ ≤ tb1− tb . (4.8.8) The last inequality follows by inverse function theorem and a estimate similar to (4.8.1). 135 Set g(y) := −(r ◦ P−1t )(y). To solve (4.8.5) we first solve the equation g(y1, y2) = 〈y1 y2 , Y t1 (y) Y t2 (y) 〉 on Dδ(1−2b). This is done by the functions Y t1 (y) = 1 y1 g(y1, 0) and Y t2 (y) = 1 y2 (g(y1, y2)− g(y1, 0)). We now show that these functions are holomorphic and derive some estimates for them. Observe that, in view of (4.8.6), the change of variables y = Pt(x) = ( x2 + t ∂r ∂x1 (x), x1 + t ∂r ∂x2 (x) ) obeys (1− 2b)(|x1|+ |x2|) ≤ |y1|+ |y2| ≤ (1 + 2b)(|x1|+ |x2|). (4.8.9) By the chain rule, (4.8.6), (4.8.8), and the last inequality, for 1 ≤ i ≤ 2,∣∣∣∣ ∂g∂yi (y) ∣∣∣∣ ≤ b(|x1|+ |x2|) ( 1 + √ 2b 1− b ) ≤ b 1 + 2b 1− 2b(|y1|+ |y2|), since √ 2b/(1− b) < 2b because b < 1/55. Therefore, |g(y1, y2)| ≤ b2 1 + 2b 1− 2b(|y1|+ |y2|) 2. This shows that Y t1 and Y t 2 are not singular at y = 0 and thus are holomorphic. Furthermore, it is easy to see that |Y t1 (y)| ≤ b 2 1 + 2b 1− 2b |y1|, ∣∣∣∣∂Y t1∂y1 (y) ∣∣∣∣ ≤ 3b2 1 + 2b1− 2b and ∂Y t2∂y1 (y) = 0. These are the estimates we need for Y t1 . We next consider Y t 2 . To derive bounds for Y t2 we consider the regions |y2| ≥ |y1| and |y2| ≤ |y1| separately. First, if |y2| ≥ |y1| then |Y t2 (y)| ≤ 1 |y2|(|g(y1, y2)|+ |g(y1, 0)|) ≤ b 1 + 2b 1− 2b (|y1|+ |y2|)2 |y2| ≤ 2b 1 + 2b 1− 2b(|y1|+ |y2|), and similarly ∣∣∣∣∂Y t2∂y1 (y) ∣∣∣∣ ≤ 1|y2| (∣∣∣∣ ∂g∂y1 (y1, y2) ∣∣∣∣+ ∣∣∣∣ ∂g∂y1 (y1, 0) ∣∣∣∣) ≤ 4b 1 + 2b1− 2b ,∣∣∣∣∂Y t2∂y2 (y) ∣∣∣∣ ≤ 1|y2| |Y t2 (y)|+ 1|y2| ∣∣∣∣ ∂g∂y2 (y1, y2) ∣∣∣∣ ≤ 6b 1 + 2b1− 2b . 136 Observe that, in particular, these estimates hold for |y1| = |y2|. Now, for fixed y1 we can apply the maximum modulus principle to derive bounds for the functions Y t2 (y1, ·) and ∂ ∂yi Y2(y1, ·) in the disk |z| ≤ |y1|. (Note, this is the case |y2| ≤ |y1|.) By this principle, the modulus of these functions inside the disk is bounded by the maximum modulus at the boundary |z| = |y1|. Thus, using the above estimates, which are valid for |y1| = |y2|, we obtain |Y t2 (y1, z)| ≤ max|z|=|y1| |Y t 2 (y1, z)| ≤ max|z|=|y1| 2b 1 + 2b 1− 2b(|y1|+ |z|) = 4b 1 + 2b 1− 2b |y1| ≤ 4b 1 + 2b 1− 2b(|y1|+ |z|). Similarly, ∣∣∣∣∂Y t2∂y1 (y1, z) ∣∣∣∣ ≤ 4b 1 + 2b1− 2b and ∣∣∣∣∂Y t2∂y2 (y1, z) ∣∣∣∣ ≤ 6b 1 + 2b1− 2b . We have all the bounds we need for Y t2 . Let Y t(y) = (Y t1 (y), Y t 2 (y)). Then, combining all the above estimates we find that, for 1 ≤ i ≤ 2 and for all y ∈ Dδ(1−2b), ‖Y t(y)‖ ≤ √ 1614 b 1 + 2b 1− 2b(|y1|+ |y2|), (4.8.10)∥∥∥∥∂Y t∂yi (y) ∥∥∥∥ ≤ 6b 1 + 2b1− 2b . (4.8.11) Furthermore, by construction the vector field Y t(y) satisfies the equation 〈y, Y t(y)〉 = −h(P−1t (y)). If we recall the change of variables y = Pt(x), this equation is equivalent to 〈Pt(x), Y t(Pt(x))〉 = −r(x). Finally, if we set Xt := Yt ◦ Pt, then Xt(x) satisfies the desired equation 〈Pt(x), Xt(x)〉 = −r(x) on P−1t (Dδ(1−2b)). Note, this is in fact equation (4.8.5). That is, we have proved (4.8.2). Furthermore, in view of (4.8.6), the region P−1t (Dδ(1−2b)) contains Dδ(1−4b). From (4.8.10), (4.8.11), (4.8.7) and (4.8.9), we obtain the estimates (4.8.3) and (4.8.4), namely, ‖Xt(x)‖ ≤ √ 1614 b (1 + 2b)2 1− 2b (|x1|+ |x2|) < 5b(|x1|+ |x2|) < 10bδ,∥∥∥∥∂Xt∂xi (x) ∥∥∥∥ ≤ 6b 1 + 2b1− 2b < 8b. This completes the proof of the theorem. 137 4.9 Appendix We now prove Propositions 4.1.2 and 4.5.4. Proposition 4.1.2 follows from part (a) of Proposition 4.5.4, which we reproduce below. Let T be a linear operator from L2C to L 2 B with B,C ⊂ Γ#, and recall the definition ‖T‖σ := max { sup b∈B ∑ c∈C |Tb,c|σ(|b− c|), sup c∈C ∑ b∈B |Tb,c|σ(|b− c|) } , where σ satisfies the hypotheses stated in p. 93. We next prove that this norm has the following properties. Proposition 4.5.4 (Properties of ‖ · ‖σ). Let S and T be linear operators from L2C to L2B with B,C ⊂ Γ#. Then: (a) ‖T‖ ≤ ‖T‖σ≡1 ≤ ‖T‖σ; (b) If B = C, then ‖S T‖σ ≤ ‖S‖σ‖T‖σ; (c) If B = C, then ‖(I + T )−1‖σ ≤ (1− ‖T‖σ)−1 if ‖T‖σ < 1; (d) |Tb,c| ≤ 1σ(|b−c|)‖T‖σ for all b ∈ B and all c ∈ C. Proof. (a) By the Cauchy-Schwarz inequality we have |(Tϕ)∧(b)| = ∣∣∣∣∣∑ c∈C Tb,c ϕ̂(c) ∣∣∣∣∣ ≤∑ c∈C |Tb,c ϕ̂(c)| = ∑ c∈C [|Tb,c|1/2 |ϕ̂(c)|] [|Tb,c|1/2] ≤ [∑ c∈C |Tb,c| |ϕ̂(c)|2 ]1/2[∑ c′∈C |Tb,c′ | ]1/2 ≤ [ sup b∈B ∑ c′∈C |Tb,c′ | ]1/2[∑ c∈C |Tb,c| |ϕ̂(c)|2 ]1/2 . Hence, ‖(Tϕ)∧‖2l2 = ∑ b∈B |(Tϕ)∧(b)|2 ≤ [ sup b∈B ∑ c′∈C |Tb,c′ | ]∑ c∈C ∑ b∈B |Tb,c| |ϕ̂(c)|2 ≤ [ sup b∈B ∑ c′∈C |Tb,c′ | ][ sup c∈C ∑ b∈B |Tb,c| ]∑ c∈C |ϕ̂(c)|2 ≤ max { sup b∈B ∑ c′∈C |Tb,c′ |, sup c∈C ∑ b∈B |Tb,c| }2 ‖ϕ̂‖l2 . 138 Thus, ‖Tϕ‖L2B ‖ϕ‖L2C = ‖(Tϕ)∧‖l2 ‖ϕ̂‖2 l2 ≤ max { sup b∈B ∑ c′∈C |Tb,c′ |, sup c∈C ∑ b∈B |Tb,c| } for all ϕ ∈ L2C . This implies that ‖T‖ ≤ max { sup b∈B ∑ c′∈C |Tb,c′ |, sup c∈C ∑ b∈B |Tb,c| } . That is, ‖T‖ ≤ ‖T‖σ≡1. This is the first inequality of part (a). The second inequality, namely, ‖T‖σ≡1 ≤ ‖T‖σ, follows immediately if we observe that σ(t) ≥ 1 for all t ≥ 0 by hypothesis (see p. 93). This proves part (a). (b) By hypothesis B = C. First observe that, since σ increases monotonically, and σ(s+ t) ≤ σ(s)σ(t) for all s, t ∈ R+, we have σ(|b− c|) = σ(|b− d+ d− c|) ≤ σ(|b− d|+ |d− c|) ≤ σ(|b− d|)σ(|d− c|) for all b, c, d ∈ B. Hence, sup b∈B ∑ c∈B |(ST )b,c|σ(|b− c|) = sup b∈B ∑ c∈B ∣∣∣∣∣∑ d∈B Sb,dTd,c ∣∣∣∣∣σ(|b− c|) ≤ sup b∈B ∑ d∈B |Sb,d|σ(|b− d|) ∑ c∈B |Td,c|σ(|d− c|) ≤ [ sup d∈B ∑ c∈B |Td,c|σ(|d− c|) ][ sup b∈B ∑ d∈B |Sb,d|σ(|b− d|) ] , and similarly we prove that sup c∈B ∑ b∈B |(ST )b,c|σ(|b− c|) ≤ [ sup c∈B ∑ d∈B |Td,c|σ(|d− c|) ][ sup d∈B ∑ b∈B |Sb,d|σ(|b− d|) ] . Thus, ‖ST‖σ ≤ ‖S‖σ‖T‖σ, as claimed. 139 (c) By hypothesis B = C. Since ‖T‖σ < 1, the Neumann series for (I + T )−1 converges (and its σ-norm is bounded). Hence, using part (b), ‖(I + T )−1‖σ = ∥∥∥∥∥ ∞∑ j=0 T j ∥∥∥∥∥ σ ≤ ∞∑ j=0 ‖T‖jσ ≤ (1− ‖T‖σ)−1, as was to be shown. (d) Since σ(t) ≥ 1 for all t ≥ 0, observing the definition of ‖ · ‖σ it is easy to see that |Tb,c| = 1 σ(|b− c|) Tb,c σ(|b− c|) ≤ ‖T‖σ for all b ∈ B and all c ∈ C. This completes the proof. We finally prove Proposition 4.1.2. Proof of Proposition 4.1.2. By applying Proposition 4.5.4(a) we find that ‖T‖ ≤ ‖T‖σ≡1 = max { sup b∈B ∑ c∈C |Tb,c|, sup c∈C ∑ b∈B |Tb,c| } . This is the desired inequality. 140 Chapter 5 Exploiting gauge invariance 5.1 A gauge transformation In this section we introduce a gauge transformation A→ A(ν) that simplify (or modify) the main estimates and results in Chapter 4. For ν ∈ {1, 2} define Ψ̂ν : Γ# → C as Ψ̂ν(b) := (1, i(−1)ν) · Â(b) i(1, i(−1)ν) · b if b 6= 0, 0 if b = 0. Observe that, for all b ∈ Γ# \ {0} and ν ∈ {1, 2}, |(1, i(−1)ν) · b| = |b1 + i(−1)νb2| = |b| ≥ 2Λ and |Ψ̂ν(b)| = |(1, i(−1) ν) · Â(b)| |b1 + i(−1)νb2| = |(1, i(−1)ν) · Â(b)| |b| ≤ √ 2 |Â(b)| |b| . Thus, the function Ψ̂ν is well-defined and for any β ≥ 0 we have ‖|b|1+βΨ̂ν(b)‖l1 ≤ √ 2 ‖|b|βÂ(b)‖l1 . Now set A(ν) := A−∇Ψν where Ψν := (Ψ̂ν)∨, 141 and to simplify the notation write ζ := (1, i(−1)ν). Since without loss of generality we have assumed that Â(0) = 0, it follows that Â(ν)(0) = 0. Furthermore, for all b ∈ Γ# \ {0}, Â(ν)(b) = Â(b)− ibΨ̂ν(b) = Â(b)− b ζ · Â(b) ζ · b = (ζ · b)Â(b)− (ζ · Â(b))b ζ · b = 1 ζ · b ( (ζ1b1 + ζ2b2)Â1(b)− (ζ1Â1(b) + ζ2Â2(b))b1, (ζ1b1 + ζ2b2)Â2(b)− (ζ1Â1(b) + ζ2Â2(b))b2 ) = 1 ζ · b ( (b2Â1(b)− b1Â2(b))ζ2, (b2Â1(b)− b1Â2(b))(−ζ1) ) = 1 ζ · b(b2,−b1) · (Â1(b), Â2(b)) (ζ2,−ζ1) = b⊥ · Â(b) ζ · b ζ ⊥ = b⊥ · Â(b) (1, i(−1)ν) · b(1, i(−1) ν)⊥. (5.1.1) Hence, for all b ∈ Γ#, |Â(ν)(b)| ≤ √ 2 |Â(b)|, and for any β ≥ 0, ‖|b|βÂ(ν)(b)‖l1 ≤ √ 2 ‖|b|βÂ(b)‖l1 . The transformation A → A(ν) is useful because of the following proposition, which is a particular case of Theorem 2.6.1 (gauge invariance). Proposition 5.1.1 (Gauge invariance). Let ν ∈ {1, 2} and assume that A1, A2 ∈ C1(R2/Γ) and V ∈ C0(R2/Γ) so that Ψν ∈ C2(R2/Γ). Then, Ker(Hk(A, V )) 6= {0} if and only if Ker(Hk(A(ν), V )) 6= {0}, and consequently F̂(A, V ) = F̂(A(ν), V ). Therefore, to study the Fermi curve of (A, V ) we may replace A by A(1) or A(2). We shall exploit this property below. 142 5.2 The regular piece revisited From now on we shall make the following hypothesis. Hypothesis 5.2.1. A1, A2 ∈ C1(R2/Γ) and V ∈ C0(R2/Γ). We now describe the main simplifications (or modifications) introduced by the gauge transformation A→ A(ν). Our first observation is that the expressions for Nd′ +Dd′,d′ and Dd′,d′′ (in Proposition 4.3.1) become simpler. Proposition 5.2.1. Under hypothesis 5.2.1, let ν ∈ {1, 2} and replace A by A(ν′). Then Jd ′d′′ ν = 0 and L d′d′′ ν′ = 0 for any d ′, d′′ ∈ G and consequently Nd′ +Dd′,d′ = Jd ′ ν′w 2 ν,d′ + (1 +K d′)wν,d′zν,d′ + Ld ′ ν zν,d′ +M d′ and Dd′,d′′ = Jd ′d′′ ν′ w 2 ν,d′ +K d′d′′wν,d′zν,d′ + Ld ′d′′ ν zν,d′ +M d′d′′ . Proof. First recall from (5.1.1) that A(ν ′)(b) = b⊥ · Â(b) (1, i(−1)ν′) · b(i(−1) ν′ ,−1) for all b ∈ Γ#\{0}, and that A(ν′)(0) = 0. Now, substitute this expression into the definitions of Jd ′ ν and L d′ ν′ in Proposition 4.3.1. Then, if we observe that (−1)ν′ = −(−1)ν , and compute (1,−i(−1)ν) · (i(−1)ν′ ,−1) = i((−1)ν′ + (−1)ν) = 0, it follows easily that Jd ′ ν = 0 and L d′ ν′ = 0. Thus, by Proposition 4.3.1, Nd′ +Dd′,d′ = Jd ′ ν′w 2 ν,d′ + (1 +K d′)wν,d′zν,d′ + Ld ′ ν zν,d′ +M d′ and Dd′,d′′ = Jd ′d′′ ν′ w 2 ν,d′ +K d′d′′wν,d′zν,d′ + Ld ′d′′ ν zν,d′ +M d′d′′ , as was to be shown. 143 We now inspect the proof of Theorem 3.4.1 (the regular piece) to see what we have gained by performing this transformation. The last proposition immediately implies that the defining equation (4.6.1), namely, β1w 2 + β2z2 + (1 + β3)wz + β4w + β5z + β6 + q̂(0) = 0, is reduced to β1w 2 + (1 + β3)wz + β5z + β6 + q̂(0) = 0, because β2 = J00ν = 0 and β4 = L 00 ν′ = 0. Thus, instead of equation (4.6.3), namely, w + β(1)2 z + g = 0, we have w + g = 0, where |g(k)| ≤ C ρ . In the above equation, observe the absence of term β(1)2 z which does not decay with respect to ρ (it is only O(1)). This yields better bounds and makes the analysis simpler. In fact, Proposition 4.6.1 becomes Proposition 5.2.2. Under the hypotheses of Theorem 3.4.1 and Hypothesis 5.2.1, after the gauge transformation A→ A(ν′) we have |F (k)− w(k)| ≤ C1 ρ (a) and ∣∣∣∣ ∂F∂k1 (k)− 1 ∣∣∣∣ ≤ C2ρ , (b) where the constants C1 and C2 depend only on ε, Λ, q and A. Consequently, after replacing Proposition 4.6.1 by the above proposition, we no longer need Proposition 4.6.2, which is an improvement of Proposition 4.6.1 necessary to take care of the term β(1)2 z (which vanishes after the gauge transformation). Thus, Lemma 4.4.2 that was used to prove Proposition 4.6.2 is not necessary anymore. This simplifies the analysis. Then, using the new bounds, the proof of Theorem 3.4.1 can be carried out in exactly the same way yielding the following improved version of Theorem 3.4.1. 144 Theorem 5.2.3 (The regular piece after the transformation A→ A(ν′)). Let 0 < ε < Λ/6 and assume that A1, A2 and V satisfy Hypothesis 5.2.1 with ‖(1+b2)Â(b)‖l1(Γ#\{0}) < 2ε/63 and ‖b2q̂(b)‖l1(Γ#) < ∞. Then, after performing the transformation A → A(ν′), there is a constant ρ = ρΛ,ε,q,A such that, for ν ∈ {1, 2}, the projection pr induces a biholomorphic map between ( F̂(A, V ) ∩ Tν(0) ) \ Kρ ∪ ⋃ b∈Γ#\{0} Tb and its image in C. This image component contains{ z ∈ C ∣∣∣ 8|z| > ρ and |z + (−1)νθν(b)| > ε for all b ∈ Γ# \ {0}} and is contained in{ z ∈ C ∣∣∣∣∣ |z + (−1)νθν(b)| > ε2 for all b ∈ Γ# \ {0} } , where θν(b) = 12((−1)νb2 + ib1). Furthermore, pr−1 : Image(pr) −→ Tν(0), y 7−→ (−i(−1)νy − r(y), y), where |r(y)| ≤ C ρ and C = CΛ,ε,q,A is a constant. This theorem provides a simpler picture than Theorem 3.4.1 because here |r(y)| decays with respect to ρ. 145 Bibliography [1] S. B. Chae, Holomorphy and calculus in normed spaces, Marcel Dekker, 1985. [2] J. Feldman, The spectrum of periodic Schrödinger operators, UBC class notes. [3] J. Feldman, H. Knörrer and E. Trubowitz, Asymmetric Fermi surfaces for magnetic Schrödinger operators, Commun. Part. Diff. Eq. 26, 319-336 (2000). [4] J. Feldman, H. Knörrer and E. Trubowitz, Riemann surfaces of infinite genus, Amer. Math. Soc., 2003. [5] D. Gieseker, H. Knörrer and E. Trubowitz, The geometry of algebraic Fermi curves, Perspectives in mathematics 14, 1992. [6] H. Knörrer and E. Trubowitz, A directional compactification of the complex Bloch va- riety, Comment. Math. Helvetici 65, 114-149 (1990). [7] I. Krichever, Spectral theory of two-dimensional periodic operators and its applications, Russian Math. Surveys 44:2, 145-225 (1989). [8] P. Kuchment, Floquet theory for partial differential equations, Birkhäuser Verlag, 1993. [9] O. Madelung, Introduction to solid state theory, Springer, 1995. [10] H. McKean, Integrable systems and algebraic curves, in Global Analysis, Proceedings, 1978, LNM 755, Springer Verlag, 1979. [11] M. Reed and B. Simon, Methods of modern mathematical physics I: functional analysis, Academic Press, 1972. [12] M. Reed and B. Simon, Methods of modern mathematical physics IV: analysis of oper- ators, Academic Press, 1978. 146 [13] W. Rudin, Real and complex analysis, McGraw-Hill, 1987. [14] M. U. Schmidt, A proof of the Willmore conjecture, preprint, arXiv:math/0203224 (2002). [15] B. Simon, Trace ideals and their applications, 2nd edition, Amer. Math. Soc., 2005. [16] J. L. Taylor, Several complex variables with connections to algebraic geometry and Lie groups, Amer. Math. Soc., 2002. 147
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Asymptotics for Fermi curves of electric and magnetic periodic fields de Oliveira, Gustavo 2009
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Title | Asymptotics for Fermi curves of electric and magnetic periodic fields |
Creator |
de Oliveira, Gustavo |
Publisher | University of British Columbia |
Date Issued | 2009 |
Description | This work is concerned with some geometrical properties of (complex) Fermi curves of electric and magnetic periodic fields. These are analytic curves in C² that arise from the study of the eigenvalue problem for periodic Schroedinger operators. More specifically, we characterize a certain class of these curves in the region of C² where at least one of the coordinates has "large" imaginary part. The new results obtained in this thesis extend previous results in the absence of magnetic field to the case of "small" magnetic field. Our theorems can be used to show that generically these Fermi curves belong to a class of Riemann surfaces of infinite genus. |
Extent | 901157 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-07-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0067333 |
URI | http://hdl.handle.net/2429/11114 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2009-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
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