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Absolutely continuous spectrum for the Anderson model on trees Halasan, Florina 2009

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Absolutely Continuous Spectrum for the Anderson Model on Trees by Florina Halasan A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) December 2009 c© Florina Halasan 2009 Abstract This dissertation is an examination of the absolutely continuous spectrum for the Anderson model on different types of trees. The text is divided into four chapter: an intro- duction, two main chapters and conclusions. In Chapter 2 the existence of purely absolutely continuous spectrum is proven for the Anderson model on a Cayley tree, or Bethe lattice, of degree K. The method used, a geometric one, is based on some properties of the hyperbolic distance. It is a simplified generalization of a result for K = 3 given by R. Froese, D. Hasler and W. Spitzer. In Chapter 3 a similar result is proven for a more general tree which has vertices of degrees 2 and 3 alternating in a periodic manner. The lack of symmetry changes the analysis, making it possible to eliminate one of the steps in the proof for the Cayley tree. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 General Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Absolutely Continuous Spectrum for the Anderson Model on a Cayley Tree 8 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 The Model and the Results . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Analysis of µ2 and Proofs of Lemmas . . . . . . . . . . . . . . . . . . . . 16 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3 Absolutely Continuous Spectrum for the Anderson Model on Some Tree-like Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Outline of the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Proofs of Lemma 3.4 and Lemma 3.5 . . . . . . . . . . . . . . . . . . . . 36 3.4 Additional Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.4.1 A Criterion for Absolutely Continuous Spectrum . . . . . . . . . . 48 3.4.2 Bounds on the Green Function at an Arbitrary Site . . . . . . . . . 49 3.5 On a recursion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 iii Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 iv List of Figures 3.1 The T tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 The nodes in the recurrence relation for the forward Green function. . . . . 32 3.3 Rearrangement of a tree. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.1 The stacked tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 v Acknowledgements This thesis was completed under the supervision of Richard Froese. Richard has been a wonderful advisor. His knowledge, guidance and friendship have been invaluable for the past four years. Thank you, Richard! Many thanks to Richard Froese, Peter Hislop, David Brydges and Joel Feldman for reading this text and giving me important suggestions. My studies were supported in part by the Natural Sciences and Engineering Re- search Council and by the Mathematics of Information Technology and Complex Systems through a graduate research fellowship. vi Chapter 1 Introduction 1.1 Preliminaries In quantum mechanics, the Schrödinger equation describes the change in the quan- tum state of a physical system. In the standard interpretation of quantum mechanics, the quantum state, also called a wave function or state vector, ψt, is the most complete descrip- tion that can be given to a physical system. Solutions to Schrödinger’s equation describe not only atomic and subatomic systems, atoms and electrons, but also macroscopic sys- tems, possibly even the whole universe. The equation of a general quantum system is i ∂tψt = H ψt, where H, the Hamiltonian, is a self-adjoint operator on a Hilbert space. We know, due to functional calculus, that its solution with initial condition ψ0 is ψt = e−itHψ0. The total energy of a particle, in quantum mechanics, is expressed as the sum of operators corresponding to the kinetic and potential energies, in the form H = T + V on the Hilbert space L2(Rn). For such a system the kinetic term is the unbounded operator T = −1 2 ∆ = 1 2 p2, with p = i∇. The potential V is a multiplication operator with a function on the configuration space, a function that depends on the application we want to consider. For random Schrödinger operators this function is a random variable. The spectral analysis of H helps us determine some physical properties of the sys- tem. The spectrum, σ, of an operator has three components. Those components are pure point spectrum, σpp, singular continuous spectrum, σsc and absolutely continuous spec- trum, σac. The pure point spectrum corresponds to energy levels for which the system is generally (depending also on other properties of the model) an insulator and the absolutely continuous spectrum corresponds to energy levels for which the system is a conductor. The study of random Schrödinger operators is an area of very active research in mathematical physics and mathematics. Since the replacement of a continuous system by a discrete one is a common approximation the physics literature, special attention has been give to the study of random Schrödinger operators on the discrete space Zd with d = 1, 2, . . ., the associated Hilbert space being l2(Zd). The ensemble of Hamiltonians of the form: Hq = ∆+k q(x), where ∆ψ(x) = ∑ |y−x|=1 ψ(y) for ψ ∈ l2(Zd), k ≥ 0 and {q(x)}x∈Zd are independent 1 identically distributed random variables is known in the literature as the Anderson model [2]. The assumptions on the random variables are not well motivated, but they are useful for simplicity. It is certainly interesting (and in many cases a challenging problem) to relax these assumptions. In his paper, Absence of diffusion in certain random lattices (1958), P. Anderson discovered one of the most striking quantum interference phenomena: particle localization due to disorder. Cited in 1977 for the Nobel prize in physics, the paper was fundamental for many subsequent developments in condensed matter theory. In particular, in the last 25 years the phenomenon of localization proved to be crucial for the understanding of the Quantum Hall effect, mesoscopic fluctuations in small conductors as well as some aspects of quantum chaotic behaviour. Random Schrödinger operators are an area of very active research in mathematical physics and mathematics. Here the main effort is to clarify the nature of the underlying spectrum. We will give a short presentation of both what physicists concluded and what mathematicians proved. There is a qualitative difference between one dimensional disordered systems (d = 1) and higher dimensional ones (d ≥ 3). For one dimensional disordered systems one expects that the whole spectrum is pure point. Thus, there is a complete system of eigen- functions which decay exponentially at infinity. This phenomenon is called Anderson lo- calization or exponential localization and it corresponds to low mobility of the electrons in our system. Therefore, one dimensional disordered systems (e.g. thin wires with impuri- ties) should have low or even vanishing conductivity and an arbitrarily small disorder will change the total spectrum from absolutely continuous to pure point. This has been already proved for all energies for an ample class of disordered systems. The physicists seem to believe that we have complete Anderson localization for d = 2 similar to the case d = 1. However, the pure point spectrum is expected to be less stable for d = 2. There are no mathematical proofs. In dimension (d ≥ 3) the physics of the system is more complicated. For small randomness, Anderson localization occurs near the band edges of the spectrum. Near any band edge a there is an inteval [a, a + δ], (respectively [a− δ, a]) of pure point spectrum and the corresponding eigenfunctions are exponentially localized in the sense that they decay exponentially fast at infinity. Inside the bands the spectrum is expected to be absolutely continuous at small disorder. Since the corresponding (generalized) eigenfunctions are cer- tainly not square integrable, one speaks of extended states or Anderson delocalization in this regime. The pure point spectrum expands when the randomness increases and the absolutely continuous part is expected to become smaller and smaller. The physics commu- 2 nity believes in the existence of a phase transition from an insulating phase to a conducting phase. A transition point between these phases is called a mobility edge. At a certain level of disorder the absolutely continuous spectrum would disappear and we will be left with only pure point spectrum. There are no mathematical results on the Anderson delocalization in `2(Zd), d ≥ 2. There is no proof of existence of absolutely continuous spectrum for any of the models we have discussed so far. In particular it is not known whether there is a conducting phase or a mobility edge at all. However, there are results for continuum models on L2(Rd), for d ≥ 2. On L2(R2), A. Klein, O. Lenoble and P. Mueller proved the existence of dynamical delocalization at energies in the Landau bands of the randomly perturbed Landau Hamil- tonian. Also, existence of absolutely continuous spectrum is known for trees. Anderson delocalization on trees is the focus of this disertation. The first result on delocalization was obtained by Abel Klein in 1998. He proved the existence of purely absolutely continuous spectrum, under weak disorder, on the Bethe lattice (or Cayley tree). This is a graph (“lattice”) without loops (hence a tree) with a fixed number of edges at every site, its degree. One considers the graph Laplacian on the Bethe lattice, which is analogously defined to the Laplacian on Zd and an independently identi- cally distributed potential on the sites of the graph. Using supersymmetric representations Klein showed that for small disorder the almost sure spectrum is purely absolutely contin- uous in an energy range which is contained in σac(∆) as shown in [6]. Moreover, the states in this energy range exhibit super-ballistic-transport behaviour [7]. Later, in 2005, Aizenman, Sims and Warzel presented a different method for estab- lishing the persistence of some absolutely continuous spectrum under weak disorder. Their work does not address the question whether the ac spectrum is pure in the intervals under study. However, the results apply to more general situations. During the same year, Froese, Hasler and Spitzer introduced a geometric method which proved the existence of purely absolutely continuous spectrum on a Bethe lattice of degree 3. Their result is similar to the one obtained by Klein in 1998, but it is only for a Bethe lattice of degree 3 whereas Klein’s is for any degree K ≥ 3. The first paper of this manuscript brings a few simplifications to the method and generalizes it to the Bethe lattice of degree K. The second paper extends the method to a more general, less symmetrical, type of trees. The model in Chapter 2 is ergodic and therefore the spectral components are almost surely constant; the one in Chapter 3 is not but nevertheless, the absolutely continuous spectrum is identified almost surely. 3 1.2 General Outline In both cases, the proofs follow some common steps; these steps will be explained in this section. Let us consider a tree T; for simplicity we will use the symbol T for both our tree and its set of vertices. We denote by o its origin. For each x ∈ T we have at most one neighbor towards the root and one or more in what we refer to as the forward direction. We say that y ∈ T is in the future of x ∈ T if the path connecting y and the root runs through x. The subtree consisting of all the vertices in the future of x, with x regarded as its root, is denoted by Tx. The Anderson Hamiltonian, H, on the Hilbert space `2(T) = { ϕ : T → C ; ∑ x∈T |ϕ(x)|2 < ∞ } is the operator of the form H = ∆ + k q where: 1. The free Laplacian ∆ is defined by (∆ϕ)(x) = ∑ y:d(x,y)=1 (ϕ(x) − ϕ(y)) , for allϕ ∈ `2(T) , with the distance d denoting the number of edges in the shortest (only) path between sites. 2. The operator q is a random potential, (qϕ)(x) = q(x)ϕ(x), where {q(x)}x∈T is a family of independent, identically distributed real random vari- ables with common probability distribution ν. We assume the 2(1 + p) moment,∫ |q|2(1+p)dν, is finite for some p > 0. The coupling constant k measures the disorder. The goal is to prove the existence of absolutely continuous spectrum for this op- erator when k is small, more precisely, that there are intervals on which all the spectral measures associated to this Hamiltonian are absolutely continuous. The analysis herein focuses on the resolvent (H − λ)−1 for λ ∈ C. The spectrum of H, denoted by σ(H), is defined to be the set of λ ∈ C such that (H − λ)−1, the resolvent, does not exist as a bounded operator. For a self-adjoint operator this spectrum is real. The matrix elements of the resolvent will often be referred to as the Green functions and denoted by G(x,y)(λ) := 〈δx, (H − λ)−1δy〉 and Gx(λ) for the diagonal elements. Here δx is the Kronecker delta function. 4 The spectral measure µx associated to δx is absolutely continuous with respect to the Lebesgue measure on some finite interval E, if lim inf β↘0 ∫ E ∣∣∣〈δx, (H − λ)−1δx〉∣∣∣1+p dα < ∞ , for some fixed p > 0 and λ = α + i β. This result is proved, for reference, in the appendix of Chapter 3. Using Fatou’s lemma and Fubini’s theorem we have E ( lim inf β↘0 ∫ E ∣∣∣〈δx, (H − λ)−1δx〉∣∣∣1+p dα) ≤ lim inf β↘0 ∫ E E (∣∣∣〈δx, (H − λ)−1δx〉∣∣∣1+p) dα , hence, for the existence of absolutely continuous spectrum in E it suffices to prove that sup λ∈R(E,) E (∣∣∣∣〈δx, (H − λ)−1δx〉∣∣∣∣1+p) < ∞ , (1.1) where R(E, ) = {z ∈ C : Re(z) ∈ E, 0 < Im(z) ≤ } is a strip along the real axis. We first prove (1.1) at x = o, the origin of the tree and then extend it to all the other spectral measures. In (1.1) we have a supremum of the expected value of the (1 + p) power of the absolute value of a Green function. We will first prove the inequality for a weight function w, instead of the absolute value, where w(z) = |z − zλ|2 Im(z)Im(zλ) = 2 (cosh(distH(z, zλ)) − 1) . Up to constants, w(z) is the hyperbolic cosine of the hyperbolic distance from z to zλ =〈 δ0, (∆ − λ)−1δ0 〉 , the Green function at the origin for the Laplacian. We use the inequality |z| ≤ 4w(z)Im(zλ) + 2|zλ| to finish the proof of (1.1). For the free Laplacian we can determine intervals of absolutely continuous spec- trum. We then prove the persistence of this spectrum for the perturbed Laplacian on closed subintervals, E. As it can already be observed the matrix elements of the resolvent, the Green func- tions, will play an important role in this dissertation. We can derive a recursion formula for G0(λ) based on the forward Green functions. Let Hx be the restriction of H to `2(Tx). The forward Green function Gx(λ) is de- fined to be the Green function for the truncated graph, given by Gx(λ) = 〈 δx, (Hx − λ)−1δx 〉 . The recursion relation for the forward Green functions on any graph can be de- termined using Schur’s formula (see [5]). In our case, since we only have trees, the for- 5 ward Green function at some vertex x ∈ T depends only on the forward Green functions at the neighbouring sites in the future of x. For example, in the case of a binary tree if x1 and x2 are the forward neighbours of a vertex x, then Gx(λ) = φ(Gx1 ,Gx2 , λ, q) where φ(z1, z2, λ, q) = −1 z1 + z2 + λ − q . This recursion expression can be easily derived using re- solvent properties. For the free Laplacian, self similarity of the tree implies that the forward Green function at the origin is a fixed point for the transformation φ. Thus, the spectrum of the free Laplacian can be determined by calculating this fixed point. Using the above mentioned recursion formula we have, for our example, w1+p(Gx(λ)) = w1+p(φ(Gx1 ,Gx2 , λ, q). By taking expectation we obtain the probabilis- tic recursion E ( w1+p(Gx(λ)) ) = E ( w1+p(φ(Gx1 ,Gx2 , λ, q)) ) . The particular values for the forward Green functions at the vertices x, x1 and x2 are different but their probability distri- bution is the same and therefore E ( w1+p(Gx(λ)) ) = E ( w1+p(Gx1(λ)) ) = E ( w1+p(Gx2(λ)) ) . If we define µ2(z1, z2, q, λ) = 2w1+p(φ(z1, z2, λ, q)) w1+p(z1) + w1+p(z2) we have the following: E ( w1+p(Gx(λ)) ) = E ( µ2 ( Gx1(λ),Gx2(λ), q, λ ) (1 2 w1+p(Gx1(λ)) + 1 2 w1+p(Gx2(λ)) )) . Now suppose we can prove µ2(z1, z2, λ, q) ≤ (1 − )χ1(z1, z2) + χ2(z1, z2) (1.2) where χ1(z1, z2) and χ2(z1, z2) are cut-off functions and χ2(z1, z2) is supported in a region where 1 2 w1+p(z1) + 1 2 w1+p(z2) < C and  > 0 is small. Then, E ( w1+p(Gx(λ)) ) ≤ (1−)E ( w1+p(Gx(λ) ) +C. This proves (1.1). The crucial estimate for the proofs is (1.2) and the main work in the thesis consists in proving this estimate (or a similar, more complicated one in Chapter 2) for the trees considered. The intermediate steps for achieving this are different depending on the choice of tree. In Chapter 2 we look at the Bethe lattice on which all nodes look alike. All these symmetries make the contraction properties of φ less obvious and two recursion steps are needed in the analysis. The tree in Chapter 3 has a little bit less symmetry and therefore the analysis requires only one recursion step. 6 Bibliography [1] M. Aizenman, R. Sims and S. Warzel. Stability of the Absolutely Continuous Spec- trum of Random Schrödinger Operators on Tree Graphs. Prob. Theor. Rel. Fields, (136):363-394, 2006. [2] P. W. Anderson. Absence of Diffusion in Certain Random Lattices Phys. Rev., (109):1492-1505, 1958. [3] R. Froese, D. Hasler and W. Spitzer. Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs. J. Func. Anal., (230):184-221, 2006. [4] R. Froese and D. Hasler and W. Spitzer. Absolutely Continuous Spectrum for the Anderson Model on a Tree: A Geometric Proof of Klein’s Theorem. Commun. Math. Phys., (269):239-257, 2007. [5] F. Halasan. Absolutely Continuous Spectrum for the Anderson Model on a Tree-like Graph arXiv:0810.2516, 2008 [6] W. Kirsch. An Invitation to Random Schrödinger Operators Proceedings of the sum- mer school on Disordered Systems, Paris 2003 [7] A. Klein. Spreading of wave packets in the Anderson Model on the Bethe Lattice. Commun. Math. Phys., (177):755-773, 1996. [6] A. Klein. Extended States in the Anderson Model on the Bethe Lattice. Advances in Math., (133):163-184, 1998. [7] M. Reed and B. Simon. Methods of modern mathematical physics I: functional anal- ysis. Academic Press Inc., New York, second edition, 1980. [8] B. Simon. Lp norms of the Borel transform and the decomposition of measures. Pro- ceedings of the American Mathematical Society, 123(12):3749-3755, Dec. 1995. 7 Chapter 2 Absolutely Continuous Spectrum for the Anderson Model on a Cayley Tree 1 2.1 Introduction One of the most important open problems in the field of random Schrödinger operators is to prove the existence of absolutely continuous spectrum for weak disorder in the Anderson model [2] in three and higher dimensions. The first result in this direction is Abel Klein’s, for random Schrödinger operators acting on a tree, or Bethe lattice, of any degree larger than 2. Klein [6] proves that for weak disorder, almost all potentials will produce absolutely continuous spectrum. This means that there must be many potentials on a tree for which the corresponding Schrödinger operator has absolutely continuous spectrum without there being an obvious reason, such as periodicity or decrease at infinity. Later on, different other proofs were given to the same result (see [4] and [1]). The goal of this chapter is to generalize the geometric method in [4] from a Bethe lattice of degree 3 to one of any degree M + 1, with M ≥ 2. 2.2 The Model and the Results A Bethe lattice (or Cayley tree), B, is a connected infinite graph with no closed loops and a fixed degree (number of nearest neighbors) at each vertex (site or point), x. The distance between two sites x and y will be denoted by d(x, y) and is equal to the length of the shortest (only) path connecting x and y. 1A version of this chapter will be submitted for publication. Halasan, F. Absolutely Continuous Spectrum for the Anderson Model on a Cayley Tree. 8 The Anderson model on the Bethe lattice is given by the random Hamiltonian H = ∆ + k q on the Hilbert space `2(B) = {ϕ : B → C ; ∑ x∈B |ϕ(x)|2 < ∞}. The (centered) Laplacian ∆ is defined by (∆ϕ)(x) = ∑ y: d(x,y)=1 ϕ(y) and has spectrum σ(∆) = [−2√M, 2√M] . The operator q is a random potential, with q(x), x ∈ B, being independent, identically distributed real random variables with common probability distribution ν. We assume the 2(1 + p) moment, ∫ |q|2(1+p)dν, is finite for some p > 0. The coupling constant k measures the disorder. As mentioned above, the existence of purely absolutely continuous spectrum for the Anderson model on the Bethe lattice was first proved, in a different manner, by Klein in 1998. Given any closed interval E contained in the interior of the spectrum of ∆ on the Bethe lattice, he proved that for small disorder, H has purely absolutely continuous spectrum in some interval E with probability one, and its integrated density of states is continuously differentiable on the interval (he only needed a finite second moment, whereas we have a finite 2(1+p) moment in our model). We prove a similar result in this chapter. A key point is the definition of a weight function appearing in the proofs. This definition is motivated by hyperbolic geometry. Theorem 2.1. For any E, with 0 < E < 2 √ M and H defined above, there exists k(E) > 0 such that for all 0 < |k| < k(E) the spectrum of H is purely absolutely continuous in [−E, E] with probability one, i.e., we have almost surely σac ∩ [−E, E] = [−E, E] , σpp ∩ [−E, E] = ∅ , σsc ∩ [−E, E] = ∅ . Let H = {z ∈ C : Im(z) > 0} denote the complex upper half plane. For convenience, we fix an arbitrary site in B to be the origin and denote it by 0. For each x ∈ B we have at most one neighbour towards the root and two or more in what we refer to as the forward direction. We say that y ∈ B is in the future of x ∈ B if the path connecting y and the root runs through x. Let x ∈ B be an arbitrary vertex, the subtree consisting of all the vertices in the future of x, with x regarded as its root, is denoted by Bx. We will write Hx for H when restricted to Bx and set Gx(λ) = 〈δx, (Hx − λ)−1δx〉 the Green function for the truncated graph. Gx is called the forward Green function. 9 Proposition 2.2. For any λ ∈ H we have G(λ) = 〈 δ0, (H − λ)−1δ0 〉 = −  ∑ x: d(x,0)=1 Gx(λ) + λ − k q(0) −1 (2.1) and, for any site x ∈ B, Gx(λ) = −  ∑ y: d(y,x)=1, y∈Bx Gy (λ) + λ − k q(x)  −1 . (2.2) Proof. We will prove (2.1); (2.2) is proven in exactly the same way. Let us write H = H̃+Γ, where H̃ = k q(0) ⊕  ⊕ x: d(x,0)=1 Hx  is the direct sum corresponding to the decomposition B = {0} ∪ ( ⋃ x: d(x,0)=1 Bx ) . The operator Γ has matrix elements 〈δx,Γδ0〉 = 〈δ0,Γδx〉 = 1 if d(x, 0) = 1, with all other matrix elements being 0. The resolvent identity gives (H̃ − λ)−1 = (H − λ)−1 + (H̃ − λ)−1Γ (H − λ)−1 . Also, (H̃ − λ)−1 = (k q(0) − λ)−1 ⊕  ⊕ x: d(x,0)=1 (Hx − λ)−1  . Thus 〈 δ0, (H̃ − λ)−1δ0 〉 = 〈 δ0, (H − λ)−1δ0 〉 + 〈 δ0, (H̃ − λ)−1Γ (H − λ)−1δ0 〉 . Hence G (λ) = (q(0) − λ)−1 − (k q(0) − λ)−1 ∑ x: d(x,0)=1 〈 δx, (H − λ)−1δ0 〉 , (2.3) 10 The resolvent formula also implies that for each x with d(x, 0) = 1,〈 δx, (H − λ)−1δ0 〉 = −Gx (λ) G (λ) . (2.4) (2.2) follows from (2.3) and (2.4).  The recursion relation for Gx(λ) that we just proved leads us to the following trans- formation φ : HM × R × H→ H defined by φ(z1, ...zM, q, λ) = −1 z1 + ... + zM + λ − q . (2.5) It is easy to see the equivalence between (2.1) and (2.5). Let q ≡ 0. If Im(λ) > 0, the transformation z 7→ φ(z, ..., z, 0, λ) has a unique fixed point, zλ, in the upper half plane, i.e. Im(zλ) > 0 (for details see Proposition 2.1, in [3]). Explicitly, zλ = −λ 2M + 1 M √ (λ/2)2 − M , where we will always make the choice Im √ · ≥ 0 (and √a > 0 for a > 0). This fixed point as a function of λ ∈ H extends continuously onto the real axis. This extension yields, for Im(λ) = 0 and |λ| < 2√M, the fixed point zλ = − λ2M + i 2M √ 4M − λ2 , lying on an arc of the circle |z| = 1/√M. When Im(λ) = 0 and |λ| ≤ E < 2√M, the arc is strictly contained in the upper half plane. Thus, when λ lies in the strip R(E, ) = {z ∈ H : Re(z) ∈ [−E, E], 0 < Im(z) ≤ } with 0 < E < 2 √ M and  sufficiently small, Im(zλ) is bounded below and |zλ| is bounded above by a positive constant. In order to prove that the spectral measures are absolutely continuous we need to establish bounds for E(|Gx(λ)|1+p). Since zλ equals Gx(λ) for the case q ≡ 0 and any x ∈ B, in order to prove the desired bounds we will use the weight function w(z) defined by w(z) = 2 (cosh(distH(z, zλ)) − 1) = |z − zλ| 2 Im(z)Im(zλ) . (2.6) 11 Up to constants, w(z) is the hyperbolic cosine of the hyperbolic distance from z to zλ, pro- vided λ ∈ R(E, ) with 0 < E < 2√M and  sufficiently small. This notation suppresses the λ dependence. In essence, we are looking at the hyperbolic cosine of the distance between Gx(λ) for the free Laplacian and the one for the perturbed one, H. The goal is to prove that this quantity, which blows up on the boundary, stays mostly finite. To prove a bound for E(w1+p(Gx(λ))) we will need to use (2.5), more than once, to express the forward Green function as a function of the forward Green functions at future nodes. As a result, the study of the following quantity becomes needed: µ3,p(z1 . . . z2M−1, q1, q2, λ) = ∑ σ w1+p(φ(φ(zσ1 . . . zσM , q1, λ), zσM+1 . . . zσ2M−1q2, λ)) w1+p(z1) + . . . + w1+p(z2M−1) where σ are all cyclic permutations. We will state here the needed lemmas, but we will give the proofs later. Lemma 2.3. For any E, 0 < E < 2 √ M and any 0 < p < 1, there exist positive constants , η1, 0 and a compact setM ∈ H2M−1 such that µ3,p|Mc×[−η1,η1]2×R(E,0) ≤ 1 − . (2.7) HereMc denotes the complement H2M−1 \M. Lemma 2.4. For any E, 0 < E < 2 √ M and any 0 < p < 1, there exist positive constants 0, C and a compact setM ∈ H2M−1 such that µ3,p|Mc×R2×R(E,0) ≤ C(1 + 2∑ i=1 |qi|2(1+p)). (2.8) Similarly, if we define µ′3,p(z1, . . . , zM+1) = w(−(M+1∑ i=1 zi + λ − q)−1)1+p w(z1)1+p + . . . + w(zM+1)1+p , then µ′3,p|Mc×R2×R(E,0) ≤ C(1 + |q|2(1+p)) . 12 Theorem 2.5. Let x be a nearest neighbour of 0. For any E, 0 < E < 2 √ M and all 0 < p < 1, there exists k(E) > 0 such that for all 0 < |k| < k(E) we have sup λ∈R(E,) E ( w1+p(Gx(λ)) ) < ∞ . Proof. In order to prove that the above quantity is bounded we need a couple of preparatory steps. Let η1 and p be given by Lemma 2.3, and choose 0 andM that work in both Lemma 2.3 and Lemma 2.4. For (z1, . . . , z2M−1) ∈ Mc, we estimate∫ R2 µ3,p(z1, . . . , z2M−1, k q1, k q2, λ)dν(q1)dν(q2) ≤ (1 − ) ∫ [ −η1 k , η1 k ]2 dν(q1)dν(q2) + C ∫ R2\ [ −η1 k , η1 k ]2(1 + 2∑ i=1 |k qi|2(1+p)) dν(q1)dν(q2) ≤ (1 − ) + C ∫ R2\ [ −η1 k , η1 k ]2 dν(q1)dν(q2) + 2C|k|2(1+p)M2(1+p) ≤ 1 − /2 provided k is sufficiently small. Here M2(1+p) denotes the moment ∫ |q|2(1+p) dν(q). The probability distributions for G and Gx on the hyperbolic plane are defined by ρG(A) = Prob{G(λ) ∈ A} and ρ(A) = Prob{Gx(λ) ∈ A}. This implies ρ(A) = Prob{φ(z1 . . . zM, k q, λ) ∈ A} = Prob{(z1 . . . zM, k q, λ) ∈ φ−1(A)} = ∫ φ−1(A) dρ(z1) . . . dρ(zM) dν(q) = ∫ HM×R χA(φ(z1 . . . zM, k q, λ)) dρ(z1) . . . dρ(zM) dν(q) which gives us that for any bounded continuous function w(z),∫ H w(z)dρ(z) = ∫ HM×R w(φ(z1, . . . , zM, k q, λ)) dρ(z1) . . . dρ(zM) dν(q). Now we have all the ingredients needed to prove our theorem. Using the previous relation twice, for λ ∈ R(E, 0), we obtain: 13 E ( w1+p(Gx(λ)) ) = ∫ H w1+p(z) dρ(z) = ∫ HM×R w1+p(φ(z1 . . . zM, k q1, λ)) dρ(z1) . . . dρ(zM) dν(q1) = ∫ HM×R2 w1+p(φ(φ(z1 . . . zM, k q1, λ), zM+1 . . . z2M−1, k q2, λ)) dρ(z1) . . . dρ(z2M−1) dν(q1)dν(q2) = ∫ HM×R2 1 2M − 1 ∑ σ w1+p ( φ(φ(zσ1 . . . zσM , k q1, λ), zσM+1 . . . zσ2M−1 , k q2, λ)) ) dρ(z1) . . . dρ(z2M−1) dν(q1)dν(q2) = 1 2M − 1 ∫ Mc (∫ R2 µ3,p(z1 . . . z2M−1, k q1, k q2, λ) dν(q1)dν(q2) ) × (w1+p(z1) + . . . + w1+p(z2M−1)) dρ(z1) . . . dρ(z2M−1) + C ≤ (1 − /2) ∫ H w1+p(z) dρ(z) + C = (1 − /2)E ( w1+p(Gx(λ) ) + C . where C is some finite constant, only depending on the choice ofM. Note: We used the fact that∫ H w1+p(z)dρ(z) = 1 2M − 1 ∫ H2M−1 ( w1+p(z1) + . . . + w1+p(z2M−1) ) dρ(z1) . . . dρ(z2M−1) This implies that for all λ ∈ R(E, 0), E ( w1+p(Gx(λ)) ) ≤ 2C  .  Theorem 2.6. Let x ∈ B. Under the hypotheses of Theorem 2.5, sup λ∈R(E,) E (∣∣∣〈δx, (H − λ)−1δx〉∣∣∣1+p) < ∞ for some  > 0. 14 Proof. It is an immediate consequence of Theorem 2.5 and the following inequality: |z| ≤ 4Im(s) |z − s| 2 Im(z)Im(s) + 2|s| . (2.9) The inequality clearly holds for |z| ≤ 2|s|. In the complementary case, we have |z| > 2|s| and thus |z− s| ≥ ||z| − |s|| ≥ |s|, implying |z|Im(z) ≤ |z|2 ≤ 2|z− s|2 + 2|s|2 ≤ 4|z− s|2. This proves (2.9). Using (2.9) with s = zλ yields that for λ ∈ R(E, ), |z| ≤ 4w(z)+C, where C depends only on E and . To finish the proof we need to transfer the estimate from ρ to ρG and therefore prove the inequality for x = 0. By symmetry it extends to any vertex x ∈ B. In the proof of the following estimate we need the elementary fact that for z1. . . zM+1 ∈ M, w1+p  M+1∑ i=1 zi + λ − q  −1 ≤ C (1 + |q|2(1+p)). Let R denote R(E, ), then sup λ∈R E (∣∣∣∣〈δ0, (H − λ)−1δ0〉∣∣∣∣1+p) = sup λ∈R ∫ H |z|1+p dρG(z) ≤ C1 sup λ∈R ∫ H w1+p(z) dρG(z) + C2 = C1 sup λ∈R ∫ HM+1×R w1+p  M+1∑ i=1 zi + λ − k q  −1 dρ(z1) . . . dρ(zM+1) dν(q) + C2 ≤ C1 sup λ∈R ∫ Mc×R µ′3,p(z1, . . . , zM+1, k q, λ) × (w1+p(z1) + . . . + w1+p(zM+1)) dρ(z1) . . . dρ(zM+1) dν(q) + C′2 ≤ C ∫ H×R (1 + |k q|2(1+p))w1+p(z) dρ(z) dν(q) + C2 ≤ C ∫ H w1+p(z) dρ(z) + C3 = C E ( w1+p(Gx(λ)) ) + C3 ≤ C4 , where C, C1, C2,C3 and C4 are positive constants.  As it was proven in [5] (or in the next chapter), this theorem implies the main result of this chapter: Theorem 2.1. For any E, with 0 < E < 2 √ M, there exists k(E) > 0 such that for all 0 < |k| < k(E) the spectrum of H is purely absolutely continuous in [−E, E] with 15 probability one, i.e., we have almost surely σac ∩ [−E, E] = [−E, E] , σpp ∩ [−E, E] = ∅ , σsc ∩ [−E, E] = ∅ . 2.3 Analysis of µ2 and Proofs of Lemmas For the proofs of our technical lemmas we need to analyse a quantity, µ2, which will prove to play a significant role in the expression for µ3,p. We define µ2 by µ2(z1 . . . zM, q, λ) = M w(φ(z1 . . . zM, q, λ)) w(z1) + . . . + w(zM) as a function from HM\{(zλ, . . . , zλ)} × R × R → R. In this section R = R(E, ), for some 0 < E < 2 √ M and  > 0. Proposition 2.7. For all z1, . . . , zM ∈ HM\{(zλ, . . . , zλ)} and λ ∈ R, µ2(z1, . . . , zM, 0, λ) < 1 . Proof. For z, s ∈ H set c(s, z) = 2(cosh(distH(s, z)) − 1) = |s − z| 2 Im(s)Im(z) . Note that z 7→ c(s, z) is strictly convex. This can be seen for example by noting that its Hessian has strictly positive eigenvalues. Also, for s = zλ, c(zλ, z) = w(z). The transfor- mation φ′(z) = −1/(z + λ) is a hyperbolic contraction (see [3], Proposition 2.1) and since φ′(z1 + . . . + zM) = φ(z1 . . . zM, 0, λ) we have φ′(Mzλ) = zλ. This implies distH(φ′(Mzλ), φ′(z1 + . . . + zM)) < distH(Mzλ, z1 + . . . + zM)⇔ cosh(distH(φ′(Mzλ), φ′(z1 + . . . + zM))) < cosh(distH(Mzλ, z1 + . . . + zM))⇔ c(zλ, φ(z1, . . . , zM, 0, λ)) < c(Mzλ, z1 + . . . + zM) = c ( zλ, (z1 + . . . + zM) M ) ≤ 1 M M∑ i=1 c(zλ, zi) , 16 hence Mc(zλ, φ(z1, . . . , zM, 0, λ)) M∑ i=1 c(zλ, zi) < 1 Also, from Proposition 2.1 [3], if Im(λ) = 0 then φ′ is a hyperbolic isometry. Therefore c(φ′(Mzλ), φ′(z1 + . . . + zM)) = c(Mzλ, z1 + . . . + zM) = c ( zλ, z1 + . . . + zM M ) ≤ 1 M M∑ i=1 c(zλ, zi) If Im(λ) = 0, then µ2(z, . . . , z, 0, λ) = 1. If Im(λ) > 0, since φ′ is a hyperbolic contraction, µ2(z, . . . , z, 0, λ) = 1 iff z1 = . . . = zM = zλ.  Since in our lemmas we will use a compactification argument, we need to under- stand the behavior of µ2(z1, . . . , zM, q, λ) as z1,. . . ,zM approach the boundary of H and λ approaches the real axis. Thus, it is natural to introduce the compactification H M × R × R. Here R denotes the closure and H is the compactification of H obtained by adjoining the boundary at infinity. (The word compactification is not quite accurate here because of the factor R, but we will use the term nevertheless.) The boundary at infinity is defined as follows. We cover the upper half plane model of the hyperbolic plane H with the atlas A = {(Ui, ψi)i=1,2}. We have U1 = {z ∈ C : Im(z) > 0, |z| < C}, ψ1(z) = z, U2 = {z ∈ C : Im(z) > 0, |z| > C} and ψ2(z) = −1/z = w. The boundary at infinity consists of the sets {Im(z) = 0} and {Im(w) = 0} in the respective charts. The compactification H is the upper half plane with the boundary at infinity adjoined. We will use i∞ to denote the point where w = 0. With this convention, µ2 is defined in the interior of the compactificationH M×R×R and we want to know how it behaves near the boundary. It turns out that in the coordinates introduced above, µ2 is a rational function. For the majority of points on the boundary the denominator does not vanish in the limit and µ2 has a continuous extension. There are, however, points where both numerator and denominator vanish and at these singular points the limiting value of µ2 depends on the direction of approach. By blowing up the singular points, it would be possible to define a compactification to which µ2 extends continuously. However, this is more than we need for our analysis. We will do a partial resolution of the singularities of µ2 and then extend µ2 to an upper semi-continuous function on the resulting compactification. 17 The reciprocal of the function w(z), χ(z) = 1 w(z) = Im(z)Im(zλ) |z − zλ|2 is a boundary defining function for H. This means that in each of the two charts above, χ is positive near infinity and vanishes exactly to first order on the boundary at infinity. Further more, we can express µ2 as follows: µ2(z1 . . . zM, q, λ) = M χ(φ(z1 . . . zM, q, λ))[ 1χ(z1) + . . . + 1 χ(zM) ] or µ2(z1 . . . zM, q, λ) = Mχ(z1) . . . χ(zM) χ(φ(z1 . . . zM, q, λ))[χ(z1) . . . χ(zM−1) + . . . + χ(z2) . . . + χ(zM)] Since χ(φ(z1 . . . zM, q, λ)) = Im(φ(z1 . . . zM, q, λ)) |zλ − φ(z1 . . . zM, q, λ)|2 = Im(z1 + . . . + zM + λ) |zλ(z1 + . . . + zM) + λzλ − qzλ + 1|2 we obtain µ2(z1 . . . zM, q, λ) = M M∏ i=1 χ(zi)|zλ M∑ i=1 zi + λzλ − qzλ + 1|2 [ M∑ j=1 M∏ i=1 i, j χ(zi)][ M∑ i=1 χ(zi)|zi − zλ|2 + Im(λ)] (2.10) We will now describe our compactification of HM × R × R. Start with HM × R × R. Our blow-up consists of writing χ(z1), . . . , χ(zM) in polar co-ordinates. Thus we introduce new variables r1 and βi and impose the equations χ(z1) = r1 β1,. . . , χ(zM) = r1βM and β21 + . . . + β 2 M = 1. The blown up space, K , is the variety in H M ×R × R ×RM+1 containing all points (z1, . . . , zM, q, λ, r1, β1, . . . , βM) that verify the blow-up constraints. The topology is the one given by the local description as a closed subset of Euclidean space. The set K\∂∞K can be identified with HM × R × R. After the first blow-up, µ2 becomes µ2 = M M∏ i=1 βi|zλ M∑ i=1 zi + λzλ − qzλ + 1|2 [ M∑ j=1 M∏ i=1 i, j βi][ M∑ i=1 βi|zi − zλ|2 + Im(λ)/r1] (2.11) 18 We can extend µ2 to an upper semi-continuous function onK by defining, for points k ∈ ∂∞K , µ2(k) = lim sup kn→k kn∈K\∂∞K µ2(kn) . Here kn = (z1,n, z2,n, . . . , zM,n, q1,n, λn) and it converges to k in K . Let us define Σ to be the subset of K where µ2 = 1 and let K0 denote the subset of ∂∞K where λ ∈ (−2 √ M, 2 √ M) and q = 0. For the analysis of µ3 we need the following lemma: Lemma 2.8. Let Γ = {k ∈ K : k = (z1, . . . , zM, 0, λ, 0, β, . . . , β)} ⊂ K , it contains points in K with β1 = . . . = βM = β. Then, Γ ∩ Σ ∩ K0 = {k ∈ K0 : k = (z, . . . , z, 0, λ, 0, β, . . . , β)} . Proof. Let us first derive an upper bound µ∗2 for µ2. For k = (z1, . . . , zM, 0, λ, r1, β1, . . . βM) ∈ K\∂∞K we have µ2(k) = M w(φ(z1, . . . , zM, q, λ)) M∑ i=1 w(zi) = M c(zλ, φ(z1, . . . , zM, q, λ)) M∑ i=1 c(zλ, zi) ≤ M c(Mzλ, z1 + . . . + zM) M∑ i=1 c(zλ, zi) = M w( 1M M∑ i=1 zi) M∑ i=1 w(zi) . Therefore we can define µ∗2(k) = M w( 1M M∑ i=1 zi) M∑ i=1 w(zi) = M∏ i=1 βi| M∑ i=1 (zi − zλ)|2 [ M∑ j=1 M∏ i=1 i, j βi][ M∑ i=1 βi|zi − zλ|2] (2.12) Clearly µ2 ≤ µ∗2, with equality when λ is real. Let k ∈ Γ ∩ Σ ∩ K0. If k is a point of continuity for µ∗2 then µ∗2(k) = 1. At a point of continuity k, 1 = µ2(k) = lim sup kn→k kn∈M\∂∞M µ2(kn) ≤ lim sup kn→k kn∈M\∂∞M µ∗2(kn) ≤ 1 . The last inequality holds because at a point of continuity, the lim sup is actually a limit 19 which can be evaluated in any order. If we take the limit in λ and q first, we may use the fact that for λ ∈ (−2√M, 2√M), µ2 = µ∗2. Proposition 2.7 proves that the limit in zi is at most 1, which implies µ∗2(k) = 1 at the points of continuity. Since we do not need to know the entire behavior of µ2 at the boundary, we will concentrate only on the situations needed in the analysis of µ3. Therefore we need two cases to consider: C I: Let k ∈ Γ ∩ Σ ∩ K0 such that z1, . . . , zM ∈ ∂∞H and zi , i∞ for all i = 1, . . . ,M. This is a point of continuity and we have: µ∗2(k) = | M∑ i=1 (zi − zλ)|2 M M∑ i=1 |zi − zλ|2 . By the triangle inequality and the Cauchy Schwarz inequality, | M∑ i=1 (zi − zλ)|2 ≤  M∑ i=1 |zi − zλ|  2 ≤ M  M∑ i=1 |zi − zλ|2  . The first inequality turns into equality if zi − zλ have the same argument for all i and the second one if zi − zλ are equal in absolute values. Therefore, µ∗2 = 1 iff all zi are equal. C II: Let k ∈ Γ ∩ Σ ∩K0, z1 = . . . = za = i∞, and za+1, . . . , zM are real, for some a, 1 < a < M. Suppose (kn) is a sequence that realizes the lim sup in the definition of µ2(k). µ∗2(kn) = | a∑ i=1 (zi − zλ) + M∑ i=a+1 (zi − zλ)|2 M M∑ i=1 |zi − zλ|2 ≤ | a∑ i=1 (zi − zλ) + M∑ i=a+1 (zi − zλ)|2 M M∑ i=1 |zi − zλ|2 . The second term in the numerator stays finite in the limit and therefore, obviously µ∗2(k) ≤ a M .  We end this section with the proofs of our previous lemmas, Lemma 2.3 and Lemma 2.4. Proof of Lemma 2.3: In order to simplify the notation, let us define Z = (z1, . . . , z2M−1), Q = (q1, q2), ξσ(Z,Q, λ) = (zσ1 , . . . , zσM , q1, λ), τσ(Z,Q, λ) = (φ(ξσ(Z,Q, λ)), zσM+1 , . . . , zσ2M−1 , q2, λ) and νi = w(zi) w(z1) + . . . + w(z2M−1) . 20 Extend µ3,p to an upper semi-continuous function on H 2M−1 × R2 × R by setting, at points Z0, Q0, λ0 where it is not already defined, µ3,p(Z0,Q0, λ0) = lim sup Z→Z0,Q→Q0,λ→λ0 µ3,p(Z,Q, λ), . The points Z, Q and λ are approaching their limits in the topology of H 2M−1 × R2 × R. To prove the lemma it is enough to show that µ3,p(Z,Q, λ) < 1 for (Z,Q, λ) in the compact set ∂∞H 2M−1 × {0}2 × [−E, E], since this implies that for some  > 0, the upper semi-continuous function µ3,p(Z,Q, λ) is bounded by 1− 2 on the set, and by 1 −  in some neighborhood. We have µ3,p(Z,Q, λ) = ∑ σ w1+p(φ(τσ(Z,Q, λ))) w1+p(z1) + . . . + w1+p(z2M−1) = ∑ σ ( w(φ(τσ(Z,Q, λ))) w(z1) + . . . + w(z2M−1) )1+p 1 ν 1+p 1 + . . . + ν 1+p 2M−1 = = ∑ σ [ µ2(τσ) ( 1 M2 µ2(ξσ)(νσ1 + . . . + νσM ) + 1 M (νσM+1 + . . . + νσ2M−1) )]1+p · · 1 ν 1+p 1 + . . . + ν 1+p 2M−1 . Define χ(z1) = 1 w(z1) = R1Ω1, . . . , χ(z2M−1) = 1 w(z2M−1) = R1Ω2M−1, where R1, Ω1, Ω2,. . . , Ω2M−1 are defined functions of Z with the property Ω21 + . . . + Ω 2 2M−1 = 1. Notice that for any cyclic permutation σ, νσl = 2M−1∏ j=1 j,l Ωσ j 2M−1∑ i=1 2M−1∏j=1 j,1 Ω j  (2.13) In the analysis of µ2(ξσ) we use the blow-up with coordinates r1σ(ξσ) and βσ j(ξσ) where j = 1, . . . ,M and in the analysis of µ2(τσ) we use the blow-up with coordinates 21 r2σ(τσ) and βσ j(τσ) where j = M, . . . , 2M − 1. Therefore we have the following relations: R1Ωσ j = r1σβσ j(ξσ) when j = 1, . . . ,M R1F = χ(φ(ξσ)) = r2σβσ1(τσ) R1Ωσ j = r2σβσ j(τσ) when j = M + 1, . . . , 2M − 1 where F = χ(φ(ξσ)) R1 = r1σM M∏ i=1 βσi R1µ2(ξσ(Z,Q, λ)) M∑ j=1 M∏ i=1 i, j βσi = MΩσ1 M∏ i=2 βσi µ2(ξσ(Z,Q, λ)) M∑ j=1 M∏ i=1 i, j βσi . Consequently Ω2σ j = β 2 σ j (ξσ)(Ω2σ1 + . . . + Ω 2 σM ) for j = 1, . . . ,M Ω2σ j = β 2 σ j (τσ)(F + Ω2σM+1 + . . . + Ω 2 σ2M−1) for j = M, . . . , 2M − 1. Suppose that µ3,p(Z,Q, λ) = 1 for some (Z,Q, λ) ∈ ∂∞H2M−1 × {0}2 × [−E, E]. Then there must exist a sequence (Zn,Qn, λn) with Zn −→ Z in H2M−1, Qn −→ (0, 0) and λn −→ λ ∈ [−E, E] such that lim µ3,p(Zn,Qn, λn) = 1. From now on Z and λ will denote the limiting values of the sequences Zn and λn. Similarly, we will denote by νi and Ωi the limits of νi(Zn) and Ωi(Zn). We claim that ν1 = . . . = ν2M−1 = 1 2M − 1 . (2.14) This follows from the expression for µ3,p(Z,Q, λ), the bound for µ2 and the convexity of x 7→ x1+p: 22 1 = µ3,p(Z,Q, λ) = ∑ σ [ µ2(τσ) ( 1 M2 µ2(ξσ)(νσ1 + . . . + νσM ) + 1 M (νσM+1 + . . . + νσ2M−1) )]1+p · · 1 ν 1+p 1 + . . . + ν 1+p 2M−1 ≤ ∑ σ [( 1 M2 (νσ1 + . . . + νσM ) + 1 M (νσM+1 + . . . + νσ2M−1) )]1+p 1 ν 1+p 1 + . . . + ν 1+p 2M−1 ≤ ∑ σ ( 1 M2 (ν1+pσ1 + . . . + ν 1+p σM ) + 1 M (ν1+pσM+1 + . . . + ν 1+p σ2M−1) ) 1 ν 1+p 1 + . . . + ν 1+p 2M−1 = 1, so the inequalities must actually be equalities. Since p > 0, strict convexity implies that equality only holds if ν1 = . . . = ν2M−1. Since their sum is 1, their common value must be 1 2M − 1. By going to a subsequence, we may assume that Ωi(Zn) converge. Then (2.13) and (2.14) imply that their limiting values along the sequence must be Ω1 = . . . = Ω2M−1 = 1√ 2M − 1 . (2.15) One consequence is that zi ∈ ∂∞H (2.16) for i = 1, . . . , 2M − 1. Now consider the values of ξσ(Zn,Qn, λn) and τσ(Zn,Qn, λn). Since these values vary in a compact region inM we may, again by going to a subsequence, assume that they converge in M to values which we will denote ξσ and τσ. Using (2.14) and the bound µ2 ≤ 1, we find that 1 = lim n→∞ ∑ σ [ µ2(τσ(Zn,Qn, λn)) ( µ2(ξσ(Zn,Qn, λn)) + M − 1 M(2M − 1) )]1+p (2M − 1)p ≤ 1 2M − 1 ∑ σ [ 1 M µ2(τσ) (µ2(ξσ) + M − 1)) ]1+p ≤ 1. This implies that for every σ occurring in the sum we have µ2(ξσ) = µ2(τσ) = 1. Therefore, using (2.16) we conclude that for each σ, ξσ and τσ lie in the set Σ given by Lemma 2.8. 23 Now consider the coordinates βσi , i = 1, . . . ,M for the point ξσ. These are the limiting values of βσi(zσ1 , . . . , zσM ) along our sequence. Since Ω 2 σ j = β2σ j(Ω 2 1 + . . . + Ω 2 M) and Ωi = 1√2M−1 , we have βσi = 1√ M for i = 1, . . . ,M. Going back to the analysis of µ2, Lemma 2.8, we conclude that the H coordinates of ξσ, namely the limiting values of zσ1 , . . . , zσM must be equal. Since this is true for every cyclic permutation, we conclude that z = z1 = z2 = . . . = z2M−1 ∈ ∂∞H. We have two distinct cases: • If z ∈ R then φ(zσ1 , . . . , zσM , q, λ) −→ φ(z, . . . , z, 0, λ) = −1Mz+λ . From the analysis of µ2, Case I, the only way τσ = (φ(z, . . . , z, 0, λ), z, . . . , z) can lie in Σ is if φ(z, . . . , z, 0, λ) = z which would imply z = zλ and this cannot happen since zλ < ∂∞H. • If z = i∞ then φ(zσ1 , . . . , zσM , q, λ) −→ 0 therefore τσ −→ (0, i∞, . . . , i∞). Since Ω2σ j = β 2 σ j (τσ)(F + Ω2σM+1 + . . . + Ω 2 σ2M−1) for j = M, . . . , 2M − 1 and F = 1√2M−1 in the limiting case, βσ j(τσ) are equal. Going back to the analysis of µ2, Case II, we conclude that µ2(τσ) < 1. Therefore, µ3,p(Z,Q, λ) < 1.  Proof of Lemma 2.4: Each term in the sum appearing in µ3,p can be estimated w1+p(φ(· · · · · · )) w1+p(z1) + . . . + w1+p(z2M−1) = (w(z1) + . . . + w(z2M−1))1+p w1+p(z1) + . . . + w1+p(z2M−1) · · ( w(φ(· · · · · · )) w(z1) + . . . + w(z2M−1) )1+p ≤ (2M − 1)p ( w(φ(· · · · · · )) w(z1) + . . . + w(z2M−1) )1+p , where φ(· · · · · · ) denotes φ(φ(zσ1 , . . . , zσM , qσ1 , λ), zσM+1 , . . . , zσ2M−1 , qσ2 , λ). Therefore it is enough to prove w(φ(· · · · · · )) w(z1) + . . . + cd(z2M−1) ≤ C(1 + 2∑ i=1 |qi|2) . 24 Let φ(· · · ) denote φ(zσ1 , . . . , zσM , q1, λ). We have w(φ(· · · · · · )) w(z1) + . . . + w(z2M−1) = ∣∣∣∣∣∣1 + zλ ( φ(. . .) + 2M−1∑ i=M+1 zσi + λ − qσ2 )∣∣∣∣∣∣2 Im [ φ(. . .) + 2M−1∑ i=M+1 zσi + λ ] · 1 2M−1∑ i=1 |zi−zλ |2 Im(zi) = = | M∑ i=1 zσi + λ − qσ1 + zλ(−1 + ( M∑ i=1 zσi + λ − qσ1)( 2M−1∑ i=M+1 zσi + λ − qσ2))|2 Im( M∑ i=1 zσi + λ) + Im( 2M−1∑ i=M+1 zσi + λ)| M∑ i=1 zσi + λ − qσ1 |2 · 1 2M−1∑ i=1 |zi−zλ |2 Im(zi) ≤ C  1 Im( 2M−1∑ i=M+1 zσi) + | − 1 + ( 2M−1∑ i=M+1 zσi + λ − qσ2)( M∑ i=1 zσi + λ − qσ1)|2 Im( M∑ i=1 zσi) + Im( 2M−1∑ i=M+1 zσi)| M∑ i=1 zσi + λ − qσ1 |2  · 1 2M−1∑ i=1 |zi−zλ |2 Im(zi) ≤ C  1 Im( 2M−1∑ i=M+1 zσi) + 2  1 Im( M∑ i=1 zσi) + | 2M−1∑ i=M+1 zσi + λ − qσ2 |2| M∑ i=1 zσi + λ − qσ1 |2 Im( M∑ i=1 zσi) + Im( 2M−1∑ i=M+1 zσi)| M∑ i=1 zσi + λ − qσ1 |2   · · 1 2M−1∑ i=1 |zi−zλ |2 Im(zi) ≤ C  1 Im( 2M−1∑ i=M+1 zσi) + 2  1 Im( M∑ i=1 zσi) + | 2M−1∑ i=M+1 zσi + λ − qσ2 |2 Im( 2M−1∑ i=M+1 zσi)   · 1 2M−1∑ i=1 |zi−zλ |2 Im(zi) . Choose the compact setM so that 2M−1∑ i=1 |zi − zλ|2/Im(zi) ≥ C > 0 for some constant C and (z1, . . . , z2M−1) ∈ Mc. Then we can estimate each term depending on whether zσi is close to zλ. If all zσi are sufficiently close to zλ, then Im(zσi) is bounded below and |zσi | is bounded above by a constant. Thus Im  2M−1∑ i=M+1 zσi  2M−1∑ i=1 |zi − zλ|2/Im(zi) ≥ Im  2M−1∑ i=M+1 zσi C ≥ C′ > 0 , Im  M∑ i=1 zσi  2M−1∑ i=1 |zi − zλ|2/Im(zi) ≥ Im  M∑ i=1 zσi C ≥ C′ > 0 25 and ∣∣∣∣ 2M−1∑ i=M+1 zσi + λ − qσ2 ∣∣∣∣2 ≤ ∣∣∣∣ 2M−1∑ i=M+1 zσi + λ ∣∣∣∣ + |qσ2 |  2 ≤ ∣∣∣∣ 2M−1∑ i=M+1 zσi + λ ∣∣∣∣2 + 1 (|qσ2 |2 + 1) ≤  ∣∣∣∣ 2M−1∑ i=M+1 zσi ∣∣∣∣ + |λ| 2 + 1  (|qσ2 |2 + 1) ≤ ∣∣∣∣ 2M−1∑ i=M+1 zσi ∣∣∣∣2 (|λ|2 + 1) + 1 (|qσ2 |2 + 1) ≤ C ∣∣∣∣ 2M−1∑ i=M+1 zσi ∣∣∣∣2 + 1 (|qσ2 |2 + 1) ≤ C (1 + |qσ2 |2), so we are done. If all zσi are far from zλ, Im( 2M−1∑ i=M+1 zσi) 2M−1∑ i=1 |zi − zλ|2/Im(zi) ≥ 2M−1∑ i=M+1 |zσi − zλ|2 ≥ 1 M − 2 | 2M−1∑ i=M+1 (zσi − zλ)|2 ≥C(1 + | 2M−1∑ i=M+1 zσi |2) so that | 2M−1∑ i=M+1 zσi + λ − qσ2 |2 / Im( 2M−1∑ i=M+1 zσi) 2M−1∑ i=1 |zi − zλ|2/Im(zi)  ≤ C(1 + |qσ2 |2) in this case too. Also, Im( M∑ i=1 zσi) 2M−1∑ i=1 |zi − zλ|2/Im(zi) ≥ M∑ i=1 |zσi − zλ|2 ≥ C(1 + | M∑ i=1 zσi |2) . If at least one zσ j is not close to zλ for j = 1, . . . ,M, the first term is still bounded. If at least one zσ j is close to zλ for j = M + 1, . . . , 2M − 1, then the second term is finite and Im( 2M−1∑ i=M+1 zσi) 2M−1∑ i=1 |zi − zλ|2/Im(zi) ≥ C + |zσ j − zλ|2 ≥ C(C + |zσ j |2) . Therefore ∣∣∣∣ 2M−1∑ i=M+1 zσi + λ − qσ2 ∣∣∣∣2 Im ( 2M−1∑ i=M+1 zσi ) 2M−1∑ i=1 |zi − zλ|2/Im(zi) ≤ C ( C1 + |zσ j |2 ) ( 1 + |qσ2 |2 ) C2 + |zσ j |2 ≤ C ( 1 + |qσ2 |2 ) . The estimates for µ′3,p are very similar. We omit the details.  26 Bibliography [1] M. Aizenman, R. Sims and S. Warzel. Stability of the Absolutely Continuous Spec- trum of Random Schrödinger Operators on Tree Graphs. Prob. Theor. Rel. Fields, (136):363-394, 2006. [2] P. W. Anderson. Absence of Diffusion in Certain Random Lattices Phys. Rev., (109):1492-1505, 1958. [3] R. Froese, D. Hasler and W. Spitzer. Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs. J. Func. Anal., (230):184-221, 2006. [4] R. Froese, D. Hasler and W. Spitzer. Absolutely Continuous Spectrum for the Ander- son Model on a Tree: A Geometric Proof of Klein’s Theorem. Commun. Math. Phys., (269):239-257, 2007. [5] R. Froese, D. Hasler and W. Spitzer. Absolutely continuous spectrum for a ran- dom potential on a tree with strong transverse correlations and large weighted loops. arXiv:0809.4197 [math-ph], to appear in Rev. Math. Phys., 2009. [5] F. Halasan. Absolutely Continuous Spectrum for the Anderson Model on a Tree-like Graph arXiv:0810.2516, 2008 [6] A. Klein. Extended States in the Anderson Model on the Bethe Lattice. Advances in Math., (133):163-184, 1998. [7] B. Simon. Spectral analysis of rank one perturbations and applications. Mathemat- ical quantum theory. II. Schrödinger operators (Vancouver, BC, 1993) C.R.M. Proc. Lecture Notes, Amer. Math. Soc., Providence, RI, (8):109-149, 1995. [8] B. Simon. Lp norms of the Borel transform and the decomposition of measures. Pro- ceedings of the American Mathematical Society, 123(12):3749-3755, Dec. 1995. 27 Chapter 3 Absolutely Continuous Spectrum for the Anderson Model on Some Tree-like Graphs 2 3.1 Introduction Random Schrödinger Operators are used as models for disordered quantum mechanical systems. In particular, the Anderson Model was introduced to describe the motion of a quantum-mechanical electron in a crystal with impurities. For this model, the states corre- sponding to an absolutely continuous spectrum describe mobile electrons. Thus, an interval of absolutely continuous spectrum is an energy range in which the material is a conductor. An outstanding open problem, the extended states conjecture, is to prove existence of absolutely continuous spectrum for the lattice Zd with d > 2. Until now, it is only for the Bethe lattice that this has been established. A first result on the topic was obtained by A. Klein, [6], in 1998; he proved that for weak disorder, on the Bethe lattice, there exists absolutely continuous spectrum for almost all potentials. More recently, Aizenman, Sims and Warzel proved similar results for the Bethe lattice using a different method (see [1]). Their method establishes the persistence of absolutely continuous spectrum under weak disorder and also in the presence of a periodic background potential. During the same time, Froese, Hasler and Spitzer introduced a geometric method for proving the existence of absolutely continuous spectrum on graphs (see [3]). In their second paper on the topic, [4], they proved delocalization for the Bethe lattice of degree 3 using this geometric approach. In this work, we provide a version of the geometric method on a more general class of trees. 2A version of this chapter has been submitted for publication. Halasan, F. Absolutely Continuous Spectrum for the Anderson Model on Some Tree-like Graphs. 28 Statement of the Main Result We prove the existence of purely absolutely continuous spectrum for the Anderson Model on a tree-like graph, T, defined as follows (see Figure 4.1). n m n m Figure 3.1: The T tree Definition 3.1. Let B be an infinite full binary tree in which each node has degree 3 except for the origin, which has degree 2. Let us call its nodes principal nodes and denote by o its origin. For the origin and each principal node there are two edges leading away from the origin. Choose one of them and call it the top edge and call the other the bottom edge. On each top edge, we add m distinct auxiliary nodes; similarly we add n, m , n, distinct auxiliary nodes on each bottom edge. Thus we obtain the tree T which has a set of principal nodes denoted by Tp and a set of auxiliary nodes denoted by Ta. The conclusions in this paper remain valid if we start with any k-nary tree. We present the binary case for simplicity. By excluding the m = n case we break some of the symmetry in our tree; this asymmetry is used in Proposition 4, Section 3. The proof for the m = n case would constitute a generalization of the Bethe lattice proof presented in [5] and be considerably longer. Using the terminology established in [1], we will use the symbol T for both our tree graph and its set of vertices. For each x ∈ T = {o} ∪ Tp ∪ Ta we have at most one neighbor towards the root and two in what we refer to as the forward direction. We say that y ∈ T is in the future of x ∈ T if the path connecting y and the root runs through x. The subtree consisting of all the vertices in the future of x, with x regarded as its root, is denoted by Tx. The Anderson Model on T is given by the random Hamiltonian, H, on the Hilbert 29 space `2(T) = { ϕ : T→ C ; ∑ x∈T |ϕ(x)|2 < ∞ } . This operator is of the form H = ∆ + k q where: 1. The free Laplacian ∆ is defined by (∆ϕ)(x) = ∑ y:d(x,y)=1 (ϕ(x) − ϕ(y)) , for allϕ ∈ `2(T) , where the distance d denotes the number of edges between sites. 2. The operator q is a random potential, (qϕ)(x) = q(x)ϕ(x), where {q(x)}x∈T is a family of independent, identically distributed real random vari- ables with common probability distribution ν. We assume the 2(1 + p) moment,∫ |q|2(1+p)dν, is finite for some p > 0. The coupling constant k measures the disorder. Our main theorem states that the above defined Anderson model exhibits purely absolutely continuous spectrum for low disorder. Theorem 3.2. Let F be the open interior of the absolutely continuous spectrum of ∆ (this spectrum depends on m and n) with a finite set of values, S , removed. For any closed subinterval E, E ⊂ F, there exists k(E) > 0 such that for all 0 < |k| < k(E) the spectrum of H is purely absolutely continuous in E with probability one. Remarks. 1). The finite set S will be properly identified in Proposition 3.7. 2). The actual definition we use for F is F := {λ ∈ R : zλ ∈ C, Im(zλ) > 0} \ S where zλ = 〈δo, (∆ − λ)−1δo〉 (δo is the indicator function at the origin). Defined like this, F is the support of the absolutely continuous component of the spectral measure of ∆ for δo without the special values contained in S . Following the ideas in Lemma 3.9 (Section 3.4), i.e. rearranging the tree and deriving a formula for the Green function at the new origin, we can prove that the set {λ ∈ R : zλ ∈ C, Im(zλ) > 0} is, in fact, the support of the pure absolutely continuous spectrum for the Laplacian ∆. Let δx ∈ `2(T) be the indicator function supported at the site x ∈ T and let R(E, ) = {z ∈ C : Re(z) ∈ E, 0 < Im(z) ≤ } be a strip along the real axis, for E defined in the previous 30 theorem. The following theorem together with the criterion from Section 3.4 gives us the proof of Theorem 3.2. Theorem 3.3. Under the hypothesis of the previous theorem, we have sup λ∈R(E,) E (∣∣∣∣〈δx, (H − λ)−1δx〉∣∣∣∣1+p) < ∞ , for all sufficiently small p > 0, some  > 0 and all x ∈ T. Proof of Theoreom 3.2. Let us consider λ = α+iβ. Using Fatou’s lemma, Fubini’s theorem and Theorem 3.3, we obtain E ( lim inf β↘0 ∫ E ∣∣∣〈δx, (H − λ)−1δx〉∣∣∣1+p dα) ≤ lim inf β↘0 ∫ E E (∣∣∣〈δx, (H − λ)−1δx〉∣∣∣1+p) dα < ∞ . Therefore we must have lim inf β↘0 ∫ E ∣∣∣〈δx, (H − λ)−1δx〉∣∣∣1+p dα < ∞ , with probability one. Since 〈δx, (H − λ)−1δx〉 is the Stieltjes transform of the measure dµx, it follows from Proposition 3.8, Section 5 that the restriction of µx to E is purely absolutely continuous with probability one. In other words, the spectral measure for H corresponding to δx, for any x ∈ T, is purely absolutely continuous in E with probability one. Therefore the operator H has purely absolutely continuous spectrum on E.  3.2 Outline of the Proof Let Gx(λ) = 〈 δx, (H − λ)−1δx 〉 denote the diagonal matrix element of the resolvent at some arbitrary vertex x ∈ T, often referred to as the Green function. Our goal is to find bounds for these Green functions. We first do so for Go(λ) and then extend the bound to all diagonal terms. Let Hx be the restriction of H to `2(Tx). The forward Green function Gx(λ) is defined to be the Green function for the truncated graph, given by Gx(λ) = 〈 δx, (Hx − λ)−1δx 〉 . 31 x1 x2 q1 qn qn+1 qn+mqM x Figure 3.2: The nodes in the recurrence relation for the forward Green function. The forward Green function Gx(λ), for x a principal node, can be expressed recurrently as a function depending on the forward Green function for the two forward principal nodes and all the random potentials in between. Thus the recurrence relation, which can be derived using resolvent properties, has the form Gx(λ) = φ(Gx1(λ),Gx2(λ), q1 . . . qM, λ), (3.1) where φ : H2 × RM × H→ H is defined by φ(z1, z2, q1 . . . qM, λ) = −1 φn(z1, q1 . . . qn, λ) + φm(z2, qn+1 . . . qn+m, λ) + λ − qM (3.2) with φ0(z, λ) = z φ1(z, q1, λ) = −1 z + λ − q1 + 1 . . . φn(z, q1 . . . qn, λ) = −1 φn−1(z, q1 . . . qn−1, λ) + λ − qn + 1 and M = m + n + 1. The nodes x, x1, x2 and the potentials q1, · · · , qM involved in the recurrence (3.1) are shown in Figure 3.2. Because the origin has degree 2, the recurrence relation for Go(λ) is given by Go(λ) = φ(Gx1(λ),Gx2(λ), q1 . . . qM, λ + 1). In the above definition H = {z ∈ C : Im(z) > 0} is the complex upper half plane. 32 Notice that since H is random, at each x ∈ T, the forward Green function Gx(λ) is an H- valued random variable. We notice that since the random potential is i.i.d., at any x ∈ Tp, Gx(λ) has the same probability distribution denoted by ρ. The Green function at the origin, Go(λ) = Go(λ), has probability distribution denoted by ρo. The transformations φn and φm, in the recursion formula, are compositions of frac- tional linear transformations, hence fractional linear transformations themselves. This im- plies φ is a rational function whose numerator and denominator have degree 2. If Im(λ) > 0, the map z 7→ φ(z, z, 0, . . . , 0, λ) is an analytic map from H to H, a hyperbolic contraction. Let zλ denote its unique fixed point in the upper half plane, a solution to the cubic equa- tion z = φ(z, z, 0, . . . , 0, λ). The set {λ : Im(λ) = 0 and zλ ∈ H} is a reunion of two disjoint open intervals on the real axis. Thus, for q ≡ 0, Gx(λ) = zλ for all x ∈ Tp and Go(λ) = φ(zλ, zλ, 0 . . . 0, λ + 1). The map H 3 λ 7→ zλ extends continuously onto the real axis. Therefore we define F := {λ : Im(λ) = 0 and zλ ∈ H} \ S where, as mentioned before, S is defined in Proposition 3.7. We should note again that the set F ∪ S is the support of the absolutely continuous component of the spectral measure for δo, for the free Laplacian. The set {zλ}λ∈E is a compact curve strictly contained in H. Thus, when λ lies in the strip R(E, ) = {z ∈ H : Re(z) ∈ E, 0 < Im(z) ≤ } with E ⊂ F closed and  sufficiently small, Im(zλ) is bounded below and |zλ| is bounded above. To prove absolutely continuous spectrum, we need the bound on |Gx|1+p stated in Theorem 2. To get this bound we first prove that w1+p(Gx) is bounded, where w is a weight function defined as follows: w(z) = 2(cosh(distH(z, zλ)) − 1) = |z − zλ| 2 Im(z)Im(zλ) . Up to constants, w(z) is the hyperbolic cosine of the hyperbolic distance from z to zλ, the Green function at the root for ∆. We have dropped the λ-dependence from the notation. Our proof relies on a pair of lemmas about the following quantity: µp(z1, z2, q1 . . . qM, λ) = w1+p(φ(z1, z2, q1 . . . qM, λ)) + w1+p(φ(z2, z1, q1 . . . qM, λ)) w1+p(z1) + w1+p(z2) , for z1, z2 ∈ H2, q1, . . . , qM ∈ R and λ ∈ R(E, ). Lemma 3.4. For any closed subinterval E, E ⊂ F and all sufficiently small 0 < p < 1, 33 there exist positive constants , η1, 0 and a compact set K ⊂ H2 such that µp|Kc×[−η1,η1]M×R(E,0)(z1, z2, q1 . . . qM, λ) ≤ 1 − . (3.3) Here Kc denotes the complement H2 \ K . Lemma 3.5. For any closed subinterval E, E ⊂ F and any 0 < p < 1, there exist positive constants 0, C and a compact set K ⊂ H2 such that µp|Kc×RM×R(E,0)(z1, z2, q1 . . . qM, λ) ≤ C M∏ i=1 (1 + |qi|2(1+p)) . (3.4) Given these two lemmas we can prove that the decay of the probability distribution function of the forward Green function at infinity is preserved as Im(λ) becomes small, provided that ν has a finite moment of order 2(1 + p). Using Lemma 3.4 and Lemma 3.5 we prove Theorem 3.6 below, the last ingredient needed in the proof. Theorem 3.6. For any closed subinterval E, E ⊂ F, there exists k(E) > 0 such that for all 0 < |k| < k(E) we have sup λ∈R(E,) E ( w1+p(Gx(λ)) ) < ∞ , for all x ∈ Tp. Proof. Let η1 and p be given by Lemma 3.4, and choose 0 andK that work in both Lemma 3.4 and Lemma 3.5. For any (z1, z2) ∈ Kc and λ ∈ R(E, ), we estimate∫ RM µp(z1, z2, kq1 . . . kqM, λ)dν(q1) . . . dν(qM) ≤ (1 − ) ∫ [− η1k , η1 k ] M dν(q1) . . . dν(qM) + C ∫ RM\[− η1k , η1 k ] M M∏ i=1 ( 1 + |k qi|2(1+p) ) dν(q1) . . . dν(qM) ≤ 1 − /2, provided |k| is sufficiently small. The probability distributions on the hyperbolic plane are defined by ρ(A) = Prob{Gx(λ) ∈ A} , 34 where x is any site in Tp. The recursion formula for the Green function implies that the distributions dρ are related by ρ(A) = Prob{φ(z1, z2, k q1 . . . k qM, λ) ∈ A} = Prob{(z1, z2, k q1 . . . k qM, λ) ∈ φ−1(A)} = ∫ φ−1(A) dρ(z1) dρ(z2) dν(q1) . . . dν(qM)) = ∫ H2×RM χA(φ(z1, z2, k q1 . . . k qM, λ)) dρ(z1) dρ(z2) dν(q1) . . . dν(qM) which gives us that for any bounded continuous function f (z)∫ H f (z)dρ(z) = ∫ H2×RM f (φ(z1, z2, k q1 . . . k qM, λ)) dρ(z1) dρ(z2) dν(q1) . . . dν(qM). Using this relation, for λ ∈ R(E, 0), we obtain E ( w1+p(Gx(λ)) ) = ∫ H w1+p(z) dρ(z) = ∫ H2×RM w1+p(φ(z1, z2, k q1 . . . k qM, λ)) dρ(z1) dρ(z2) dν(q1) . . . dν(qM) = ∫ H2×RM 1 2 ( w1+p(φ(z1, z2, k q1 . . . k qM, λ)) + w1+p(φ(z2, z1, k q1 . . . k qM, λ)) ) dρ(z1) dρ(z2) dν(q1) . . . dν(qM) = 1 2 ∫ Kc (∫ RM µp(z1, z2, k q1 . . . k qM, λ) dν(q1) . . . dν(qM) ) × (w1+p(z1) + w1+p(z2)) dρ(z1) dρ(z2) + C ≤ (1 − /2) ∫ H w1+p(z) dρ(z) + C = (1 − /2)E ( w1+p(Gx(λ)) ) + C, where C is some finite constant, only depending on the choice of K . This implies that for all λ ∈ R(E, 0), E ( w1+p(Gx(λ)) ) ≤ 2C  .  Proof of Theorem 2. It is an immediate consequence of Theorem 3.6, Lemma 3.9 and the following inequality which holds for any two complex numbers z and s in H: |z| ≤ 4Im(s) |z − s| 2 Im(z)Im(s) + 2|s| . (3.5) 35 The inequality clearly holds for |z| ≤ 2|s|. In the complementary case, we have |z| > 2|s| and thus |z − s| ≥ ||z| − |s|| ≥ |s|, implying |z|Im(z) ≤ |z|2 ≤ 2 ( |z − s|2 + |s|2 ) ≤ 4|z − s|2 and further |z| ≤ 4|z − s|2/Im(z). This proves (3.5). By using (3.5) with s = zλ we obtain that for λ ∈ R(E, ) |z| ≤ 4w(z) + C , where C depends only on E and . Lemma 3.9 extends Theorem 3.6 to all x ∈ T and due to the previous inequality the statement of Theorem 3.3 follows.  3.3 Proofs of Lemma 3.4 and Lemma 3.5 In this section we will prove the bounds for µp stated in Lemma 3.4 and Lemma 3.5. In order to do so we extend µp, define some quantities to simplify the calculations and prove Proposition 3.7. We prove Lemma 3.4 with the use of Proposition 3.7 and then prove Lemma 3.5. Since in our lemmas we will use a compactification argument, we need to under- stand the behavior of µp(z1, z2, q1 . . . qM, λ) as z1, z2 approach the boundary of H and λ approaches the real axis. Thus, it is natural to introduce the compactification H 2 × RM × R. Here R denotes the closure and H is the compactification of H obtained by adjoining the boundary at infinity. (The word compactification is not quite accurate here because of the factor R, but we will use the term nevertheless.) The boundary at infinity is defined as follows. We cover the upper half plane model of the hyperbolic plane H with the atlasA = {(Ui, ψi)i=1,2}. We have U1 = {z ∈ C : Im(z) > 0, |z| < C}, ψ1(z) = z, U2 = {z ∈ C : Im(z) > 0, |z| > C} and ψ2(z) = −1/z = u. The boundary at infinity consists of the sets {Im(z) = 0} and {Im(u) = 0} in the respective charts. The compactification H is the upper half plane with the boundary at infinity adjoined. We will use i∞ to denote the point where u = 0. We defined µp for z1, z2 ∈ H2 and λ ∈ R(E, ), and now we extend µp to an upper 36 semi-continuous function on H 2 × RM × R by defining it as µp(z1,0, z2,0, q1 . . . qM, λ0) = lim sup z1→z1,0,z2→z2,0,λ→λ0 µp(z1, z2, q1 . . . qM, λ) , at points (z1,0, z2,0) and λ0 where it is not already defined. Here, the points (z1, z2) and λ are approaching their limits in the topology of H 2 × R. For computational purposes we define the following quantities: An = (1 + λ − q1 + z1)(1 + λ − q2 + φ1(z1, q1, λ)) . . . (1 + λ − qn + φn−1(z1, q1 . . . qn−1, λ)) Am = (1 + λ − qn+1 + z1)(1 + λ − qn+2 + φ1(z1, qn+1, λ)) . . . (1 + λ − qM−1 + φm−1(z1, qn+1 . . . qM−2, λ)) Cn = (1 + λ + zλ)(1 + λ + φ1(zλ, 0 . . . 0, λ)) . . . (1 + λ + φn−1(zλ, 0 . . . 0, λ)) and similarly, if we replace z1 with z2 we obtain Bn and Bm. Cm is defined analogously to Cn with m factors in the product instead of n. If we expand the expressions for An, Bn and Cn, respectively Am, Bm and Cm, defined above we can see that they are linear polynomials in the zi variable (i can be 1, 2 or λ). It is also worth mentioning that Cn , 0, respectively Cm , 0, and if Im(zi) > 0 then An, Am, Bn, Bm are also different from 0. For more properties of these quantities see Section 3.5. For the proof of Lemma 3.4 we need the following result: Proposition 3.7. For all z1, z2 ∈ ∂∞H2 and λ ∈ E, µ0(z1, z2, 0 . . . 0, λ) < 1 . (3.6) Here E is any closed interval with E ⊂ int(F \ S ). Remark. In the case m = n, µ0 is symmetric in z1 and z2 and equals 1 at some points on the boundary. To then prove our desired result we would need to go back one more step in our recurrence formula and analyse a more complicated version of µp. Proof of Proposition 3.7. Let us assume n > m. For z1, z2 ∈ H2\(zλ, zλ) we write z1 = x1+iy1 and z2 = x2 + iy2. Using these conventions, the triangle inequality and some simplifications 37 we have w(φ(z1, z2, 0 . . . 0, λ)) = |(z1 − zλ)(BmCm) + (z2 − zλ)(AnCn)|2 (y1|Bm|2 + y2|An|2)(|Cm|2 + |Cn|2)Im(zλ) (3.7) ≤ (|z1 − zλ||BmCm| + |z2 − zλ||AnCn|) 2 (y1|Bm|2 + y2|An|2)(|Cm|2 + |Cn|2)Im(zλ) and a similar inequality for w(φ(z2, z1, 0 . . . 0, λ)). These inequalities give us µ0(z1, z2, 0 . . . 0, λ) ≤ N/D (3.8) with N = ( (|z1 − zλ||BmCm| + |z2 − zλ||AnCn|)2(y2|Am|2 + y1|Bn|2)+ (3.9) (|z2 − zλ||AmCm| + |z1 − zλ||BnCn|)2(y1|Bm|2 + y2|An|2) ) y1y2 and D = (|Cm|2 + |Cn|2)(y2|Am|2 + y1|Bn|2)(y1|Bm|2 + y2|An|2) (3.10) (|z1−zλ|2y2 + |z2 − zλ|2y1) . It is easy to check that N/D ≤ 1 for z1, z2 ∈ H2 \ (zλ, zλ), but we do not need this since the statement of our proposition only refers to the boundary ∂(H 2 ) = ∂(H)×∂(H)∪∂(H)×H∪H×∂(H) where ∂(H) = R∪{i∞}. We know µ0 ≤ N/D ≤ 1, so we need to prove that at least one inequality is strict on the boundary. A few cases are to be considered: Case I: Both z1 and z2 are on the real axis. Let (z1,i, z2,i, λi) → (z1, z2, λ) be a sequence that realizes the lim sup in the definition of µ0. Notice that since y1,i → 0 and y2,i → 0, lim i→∞N = limi→∞D = 0 so the limit of N/D may depend on the direction in which z1,i and z2,i approach z1and z2 . All the following variables will in fact be sequences determined by (z1,i, z2,i, λi). We will sometimes suppress the index i for simplicity. In order to deal with this undetermined case we use a blow-up, more precisely we write y1 and y2 in the following form: y1 = r1ω1 y2 = r1ω2 38 with ω21 + ω 2 2 = 1 and r1 > 0, all functions of z1 and z2. By going to a subsequence if needed, assume ω1 and ω2 converge as i → ∞. After cancelling a factor r1, N and D in (3.8) become N = ( (|z1 − zλ||BmCm| + |z2 − zλ||AnCn|)2(ω2|Am|2 + ω1|Bn|2)+ (3.11) (|z2 − zλ||AmCm| + |z1 − zλ||BnCn|)2(ω1|Bm|2+ω2|An|2) ) ω1ω2 , D = (|Cm|2 + |Cn|2)(ω2|Am|2 + ω1|Bn|2)(ω1|Bm|2 + ω2|An|2) (3.12) (|z1 − zλ|2ω2 + |z2 − zλ|2ω1) . Let us first look at the points on the boundary where D has a non vanishing limit. The points where D→ 0 will need extra blow-ups and will be analysed afterwards. We first show that N/D ≤ 1 which is equivalent to proving the polynomial P(X,Y) =X2ω2 ( ω1ω2|AmBmCn|2 + ω1ω2|AnBnCm|2 + ω22|AnAm|2(|Cm|2 + |Cn|2) ) + Y2ω1 ( ω1ω2|AmBmCn|2 + ω1ω2|AnBnCm|2 + ω21|BnBm|2(|Cm|2 + |Cn|2) ) − 2XYω1ω2|CmCn| ( |AnBm|(ω2|Am|2 + ω1|Bn|2) + |AmBn|(ω1|Bm|2 + ω2|An|2) ) being positive; here X = |z1 − zλ| and Y = |z2 − zλ|. It is easy to see that P(X,Y) ≥ 0 since its discriminant has the form ( |AmBm||Cn|2 − |AnBn||Cm|2 )2 ( ω2|Am|2 + ω1|Bn|2 )( ω1|Bm|2 + ω2|An|2 ) . Let us now assume µ0 = 1, so that µ0 = N/D = 1, and prove that the number of λ values for which this can happen is finite. The condition µ0 = N/D, which means equality in (3.7), is equivalent to the existence of p1, p2, s1, s2 positive real numbers and γ and δ reals such that BmCm(z1 − zλ) = p1eiδ, AnCn(z2 − zλ) = p2eiδ, AmCm(z2 − zλ) = s1eiγ, BmCm(z1 − zλ) = s2eiγ, 39 which implies AmBmC2m(z1 − zλ)(z2 − zλ) = p1s1ei(δ+γ), AnBnC2n(z1 − zλ)(z2 − zλ) = p2s2ei(δ+γ), and therefore p1s1AnBnC2n = p2s2AmBmC 2 m. This equality can be true iff 1). both sides are 0 or 2). we have only non-zero terms which means, since An, Am, Bn, Bm are all real, (Cn/Cm)2 must be real. Let us look at each of these two scenarios in detail. 1). There are a few ways in which the right hand side of our equality can vanish. a) p1 = 0; this implies Bm = 0. Now, N/D = 1 iff the discriminant mentioned above is 0 which can happen if: – |An| = 0 which means we are in the case D = 0 discussed later; – ω2 = 0, we are again in the case D = 0, – ω1 = 0, |Am| = 0 we are in the case D = 0, – ω2 = 0, |Bn| = 0 we are in the case D = 0, – |Bn| = 0; in this case Bm = Bn = 0 and according to Lemma 3.10 this can happen for at most a finite number of λ values which will be included in S . b) s1 = 0; this implies Am = 0 and the analysis will be almost identical to the one in a). c) An = 0; this implies p2 = 0 and we are in a similar case to a). d) Bn = 0; this implies s2 = 0 and we are in a similar case to b). We should also notice Cn , 0 and Cm , 0. 2). ( Cn Cm )2 ∈ R. We have two possibilities: • Cn Cm ∈ R which according to Lemma 3.10 can be true for at most a finite number of λ values which will be included in S , • Cn Cm = r i, r ∈ R, which according to Lemma 3.10 can be true for at most a finite number of λ values which will be included in S . The points where D → 0 have to be analysed separately. There are a few ways in which our denominator can vanish. The first and the last term in the expression for D cannot be zero, it is only the two middle factors that can become 0. The following situations arise: 40 Scenario 1: (ω2|Am|2 + ω1|Bn|2) 9 0 and (ω1|Bm|2 + ω2|An|2) → 0. This situation can happen if: • ω2 → 0 and |Bm|2 → 0, or • |An|2 → 0 and ω1 → 0. Since the analysis of these two cases is almost identical, we will only look at this second one. We need to consider a blow-up: |An|2 = r2 sin(α) ω1 = r2 cos(α) with r2 > 0 and α ∈ [0, pi/2] functions of z1, z2 and λ. With this new blow-up we have N = ( (|z1 − zλ||BmCm| + |z2 − zλ|(r2 sin(α))1/2|Cn|)2(ω2|Am|2 + r2 cos(α)|Bn|2) + (|z2 − zλ||AmCm| + |z1 − zλ||BnCn|)2(r2 cos(α)|Bm|2 + ω2r2 sin(α)) ) r2 sin(α)ω2 and D = (|Cm|2 + |Cn|2)(ω2|Am|2 + r2 cos(α)|Bn|2)(r2 cos(α)|Bm|2 + ω2r2 sin(α)) · (|z1 − zλ|2ω2 + |z2 − zλ|2r2 cos(α)) . By going to a subsequence if needed we can assume that r2,i, Bm,i, Cm,i, Cn,i, ω2,i, αi converge to 0, Bm, Cm, Cn, 1, α respectively (recall that Bm,i is a linear polynomial in z2,i). In this situation we find µ0 ≤ |Bm| 2|Cm|2 cos(α) (|Cm|2 + |Cn|2)(sin(α) + |Bm|2 cos(α)) < 1 . • The last case under this scenario is |An|2 → 0 and |Bm|2 → 0. After a blow-up of the form |An|2 = r3 cos(β) |Bm|2 = r3 sin(β) 41 with r3 > 0 and β ∈ [0, pi/2] functions of z1, z2 and λ, we have N = ( (|z1 − zλ|(sin(β))1/2|Cm| + |z2 − zλ|(cos(β))1/2|Cn|)2(ω2|Am|2 + ω1|Bn|2)+ (|z2 − zλ||AmCm| + |z1 − zλ||BnCn|)2(ω1 sin(β) + ω2 cos(β)) ) ω1ω2 , D = (|Cm|2 + |Cn|2)(ω2|Am|2 + ω1|Bn|2)(ω1 sin(β) + ω2 cos(β)) (|z1 − zλ|2ω2+|z2 − zλ|2ω1) . Now, the new expression for D would vanish only if ω1 = 0 and cos(β) = 0, or sin(β) = 0 and ω2 = 0 respectively. This means we have the cases ω1 = 0 and |An|2 = 0, or |Bm|2 = 0 and ω2 = 0 which were already discussed. Otherwise, the expression N/D is well defined and by arguments similar to the ones before strictly less than unity. Scenario 2: (ω2|Am|2 + ω1|Bn|2)→ 0 and (ω1|Bm|2 + ω2|An|2) 9 0. This can happen if: • ω2 → 0 and |Bn|2 → 0, or • |Am|2 → 0 and ω1 → 0 or • |Am|2 → 0 and |Bn|2 → 0. Since this scenario is very much the same as the previous one we will not discuss it any further. Scenario 3: (ω2|Am|2 + ω1|Bn|2) → 0 and (ω1|Bm|2 + ω2|An|2) → 0. This situation can happen if: • ω2 → 0, |Bn|2 → 0 and |Bm|2 → 0, or • ω1 → 0, |An|2 → 0 and |Am|2 → 0. Again, due to the symmetry of our expression, it is enough to look at this second case. We need a blow-up of the form ω1 = r3 γ1 |An|2 = r3 γ2 |Am|2 = r3 γ3 42 with γ21 + γ 2 2 + γ 2 3 = 1 and r3 > 0, functions of z1, z2 and λ. N = ( (|z1 − zλ||BmCm| + |z2 − zλ|(r3γ2)1/2|Cn|)2(ω2γ3 + γ1|Bn|2)+ (|z2 − zλ|(r3γ2)1/2|Cm| + |z1 − zλ||BnCn|)2(γ1|Bm|2+ω2γ2) ) γ1ω2 , D = (|Cm|2 + |Cn|2)(ω2γ3 + γ1|Bn|2)(γ1|Bm|2 + ω2γ2) (|z1 − zλ|2ω2 + |z2 − zλ|2(r3γ2)1/2) . By going to a subsequence if needed we can assume that r3,i, Bm,i, Cm,i, Cn,i, ω2,i, γ1,i, γ2,i, γ3,i converge to 0, Bm, Cm, Cn, 1, γ1, γ2, γ3 respectively. In this situation we obtain µ0 ≤ ( |BmCm|2(γ3 + γ1|Bn|2) + |BnCn|2(γ1|Bm|2 + γ2) ) γ1 (|Cm|2 + |Cn|2)(γ3 + γ1|Bn|2)(γ1|Bm|2 + γ2) < 1 , provided the denominator does not vanish. If γ1 = γ2 = 0, γ1 = γ3 = 0, γ3 = |Bn|2 = 0 or/and |Bm|2 = γ2 = 0 extra blow-ups are needed, but the limiting value for N/D stays strictly less than 1. As an example, if we consider the extra blow-up given by γ3 = r4γ3, γ1 = r4γ1, γ21 + γ 2 3 = 1 and r4 > 0 we have µ0 ≤ |BnCn| 2γ1 (|Cm|2 + |Cn|2)(γ3 + γ1|Bn|2) < 1 . • |Am|2 → 0, |Bn|2 → 0, |Bm|2 → 0 and |Am|2 → 0. After a needed blow-up the expressions will look similar to the ones in (3.11) and (3.12), but in the blown-up variables. Case II: Both z1 and z2 are i∞. Let (z1, j, z2, j, λ j) ∈ H2×R be a sequence that realizes the lim sup in the definition of µ0. We sometimes suppress the index j for simplicity. We consider the change of variables, u1 = − 1z1 , u2 = − 1z2 and uλ = − 1zλ ; now, both u1 and u2 approach 0. With these new variables, N and D from (3.8) are given by N = ((|uλ − u1||u2uλBmCm| + |uλ − u2||u1uλAnCn|)2(Im(u2)|u1Am|2 + Im(u1)|u2Bn|2) + (|uλ − u2||u1uλAmCm| + |uλ − u1||u2uλBnCn|)2 · (Im(u1)|u2Bm|2 + Im(u2)|u1An|2))Im(u1)Im(u2), 43 D = ( |Cm|2 + |Cn|2 ) |uλ|2 ( Im(u2)|u1Am|2 + Im(u1)|u2Bn|2 ) · ( Im(u1)|u2Bm|2 + Im(u2)|u1An|2 ) ( |uλ − u1|2Im(u2) + |uλ − u2|2Im(u1) ) . Since both sequences u1, j and u2, j are approaching 0, we can write u1, j = r j cos(γ j)eix j and u2, j = r j sin(γ j)eiy j , with γ j, x j, y j ∈ [0, pi/2]. By going to a subsequence if needed and recalling that An, j, Am, j, Bn, j and Bm, j are linear polynomial is z1 and z2, we can assume that r j, u1, jAn, j, u1, jAm, j, u2, jBn, j, u2, jBm, j, uλ, jCn, j, uλ, jCm, j, γ j, x j, y j converge to 0, An, Am, Bn, Bm, Cn, Cm, γ, x, y respectively. After cancelling the common factor of r j and |uλ, j| in the above expressions for N and D and taking the limit we get N = ((|BmCm| + |AnCn|)2( sin(γ) sin(y)|Am|2 + cos(γ) sin(x)|Bn|2) + (|AmCm| + |BnCn|)2( sin(γ) sin(y)|An|2 + cos(γ) sin(x)|Bm|2)) · sin(x) sin(y) sin(γ) cos(γ) , D = (|Cm|2 + |Cn|2)( sin(γ) sin(y)|Am|2 + cos(γ) sin(x)|Bn|2)( sin(γ) sin(y)|An|2 + cos(γ sin(x)|Bm|2)( sin(x) cos(γ) + sin(y) sin(γ)) . If we compare this with Case I and consider |z1 − zλ| = |z2 − zλ|, y1 = cos(γ) sin(x) and y2 = sin(γ) sin(y) we can see that we are in a similar situation to the one in Case I . Case III: z1 ∈ R and z2 = i∞, respectively z2 ∈ R and z1 = i∞. We consider again a sequence that realizes the lim sup in the definition of µ0 and we use the same change of variables for z2, as before. Since z1 → R and u2 → 0 we can write u2 = reiy2 with r > 0 and Im(z1) = y1. After we cancel in both N and D a factor of r6 and |uλ|2 we have N = ((|z1 − zλ||u2BmuλCm| + |reiy2 − uλ||AnCn|)2(r sin(y2)|Am|2 + y1|u2Bn|2) (3.13) + (|reiy2 − uλ||AmuλCm| + |z1 − zλ||u2BnuλCn|)2(y1|u2Bm|2 + r sin(y2)|An|2))ry1 sin(y2), D = (|uλCm|2 + |uλCn|2) ( r sin(y2)|Am|2 + y1|u2Bn|2 ) ( y1|u2Bm|2 + r sin(y2)|An|2 ) ( |z1 − zλ|2r sin(y2)|uλ|2 + |reiy2 − uλ|2y1 ) . (3.14) If we compare it with Case I and consider |z1 − zλ| = |(z1 − zλ)uλ|, |z2 − zλ| = |reiy2 − uλ|, y1 = y1, y2 = r sin(y2), |An| = |An|, |Am| = |Am|, |Bn| = |u2Bn| and |Bm| = |u2Bm| we can see that the blow-ups needed are similar to the ones in Case I and we can conclude N/D < 1. 44 Case IV: z1 ∈ H and z2 ∈ R, respectively z2 ∈ H and z1 ∈ R. As before, we take a sequence that realizes the lim sup in the definition of µ0. If we look at the expressions for D given by (3.10) we can see that we can have the following three undetermined cases: y2 → 0 and |Bn|2 → 0, similarly y2 → 0 and |Bm|2 → 0, or y2 → 0, |Bn|2 → 0 and |Bm|2 → 0. The analysis of these blow-up cases can be done in a similar manner with the one from Case I and we can conclude that N/D is strictly less than 1. Case V: z1 ∈ H and z2 = i∞, respectively z2 ∈ H and z1 = i∞. We take a sequence that realizes the lim sup and we consider the same change of variables as in Case III. With the same notations u2 = reiy2 with r > 0 and Im(u1) = y1 we obtain the same expressions for N and D as in (3.13) and (3.14). With similar blow-ups with the ones in Case III we can conclude that also in this last case the limiting value for N/D is strictly less than 1.  Proof of Lemma 3.4. To prove the lemma it is enough to show that µp(Z,Q, λ) < 1 for (Z,Q, λ) in the compact set ∂∞(H 2 ) × {0}M × E, since this implies that for some  > 0, the upper semi-continuous function µp(Z,Q, λ) is bounded by 1−2 on the set, and by 1−  in some neighborhood. Let us rewrite µp in terms of µ0. µp(Z,Q, λ) = w1+p(φ(z1, z2, q1 . . . qM, λ)) + w1+p(φ(z2, z1, q1 . . . qM, λ)) w1+p(z1) + w1+p(z2) ≤ ( w(φ(z1, z2, q1 . . . qM, λ)) + w(φ(z2, z1, q1 . . . qM, λ)) w(z1) + w(z2) )1+p · · 1 ν 1+p 1 + ν 1+p 2 , where νi = w(zi) w(z1) + w(z2) , for i = 1, 2 . Since we are concentrating on the boundary of H 2 , we need the following blow-up χ(z1) = 1 w(z1) = R1Ω1 χ(z2) = 1 w(z2) = R1Ω2 45 where R1,Ω1 and Ω2 are defined as functions of z1 and z2 with the property Ω21 + Ω 2 2 = 1. Using the result in Proposition 3.7 we have µp|∂∞(H)2×{0}M×E < ( µ0|∂∞(H2)×{0}M×E )1+p (Ω1 + Ω2)1+p Ω 1+p 1 + Ω 1+p 2 ≤ (1 − )2p < 1 , for sufficiently small p.  Proof of Lemma 3.5. Each term in the sum appearing in µp can be estimated w1+p(φ(z1, z2, q1 . . . qM, λ)) w1+p(z1) + w1+p(z2) = (w(z1) + w(z2))1+p w1+p(z1) + w1+p(z2) ( w(φ(z1, z2, q1 . . . qM, λ)) w(z1) + w(z2) )1+p ≤ 2p ( w(φ(z1, z2, q1 . . . qM, λ)) w(z1) + w(z2) )1+p . Now it is enough to prove that w(φ(z1, z2, q1 . . . qM, λ)) w(z1) + w(z2) ≤ C M∏ i=1 (1+ |qi|2) , since this bounds each term in µp by the desired quantity. With the notations introduced at the beginning of this section, φn(z1, q1 . . . qn, λ) = −An−1An and applying Cauchy-Schwarz inequality twice we get w(φ(z1, z2, q1 . . . qM, λ)) w(z1) + w(z2) = |1 + zλφn(z1, q1 . . . qn, λ) + zλφm(z2, qn+1 . . . qn+m, λ) + zλ(λ − qM)|2 Im(φn(z1, q1 . . . qn, λ)) + Im(φm(z2, qn+1 . . . qn+m, λ)) + Im(λ) · 1 2∑ i=1 |zi−zλ |2 Im(zi) ≤ |AnBm − zλAn−1Bm − zλBm−1An + AnBmzλ(λ − qM)| 2 Im(z1)|Bm|2 + Im(z2)|An|2 · 1 2∑ i=1 |zi−zλ |2 Im(zi) ≤ (1 + |zλ|2)( |AnBm|2Im(z1)|Bm|2 + Im(z2)|An|2 + (|AnBm|2 + |An−1Bm + AnBm−1|2)·( 1 + |qM − λ|2) Im(z1)|Bm|2 + Im(z2)|An|2 ) · 1 2∑ i=1 |zi−zλ |2 Im(zi) ≤ (1 + |zλ|2) ( |An|2(2 + |qM − λ|2) Im(z1) + 2(1 + |qM − λ|2) ( |An−1|2 Im(z1) + |Bm−1|2 Im(z2) )) · 1 2∑ i=1 |zi−zλ |2 Im(zi) . 46 Since φn is a fractional linear transformation with coefficients given by the product matrix n∏ i=1 0 −11 1 + λ − qi , An, the denominator of φn, is a linear polynomial in z whose coefficients can be bounded above. We get |An| ≤ n |1 + z1| n∏ |1+λ−qi |≥1 i=1 |1 + λ − qi| which implies, |An|2 ≤ C ( 1 + |z1|2 ) n∏ i=1 ( 1 + |qi|2 ) . Going back to our inequality we have w(φ(z1, z2, q1 . . . qM, λ)) w(z1) + w(z2) ≤ C ( C1(1 + |qM |2) n∏ i=1 (1 + |qi|2)1 + |z1| 2 Im(z1) + C2(1 + |qM |2) n−1∏ i=1 (1 + |qi|2)1 + |z1| 2 Im(z1) + C3(1 + |qM |2) n+m−1∏ k=n+1 (1 + |qi|2)1 + |z2| 2 Im(z2) ) · 1 2∑ i=1 |zi−zλ |2 Im(zi) . Choose the compact set K such that 2∑ i=1 |zi − zλ|2/Im(zi) ≥ C > 0 for some constant C and (z1, z2) ∈ Kc. Then we can estimate each term depending on whether zi is close to zλ. If z j, j = 1, 2, is sufficiently close, then Im(z j) is bounded below and |z j| is bounded above by a constant. Thus Im(z j) 2∑ i=1 |zi − zλ|2/Im(zi) ≥ Im(z j)C ≥ C′ > 0 and 1 + |z j|2 ≤ C , so we are done. If z j, j = 1, 2, is far from zλ, Im(z j) 2∑ i=1 |zi − zλ|2/Im(zi) ≥ |z j − zλ|2 ≥ C(1 + |z j|2) so 1 + |z j|2/ ( Im(z j) 2∑ i=1 |zi − zλ|2/Im(zi) ) ≤ C again.  47 3.4 Additional Results This section contains two theorems on absolutely continuous spectrum. The first one gives a sufficient condition for a measure to be absolutely continuous with respect to the Lebesgue measure on an interval and the latter gives a sufficient condition for a random Schrödinger operator to exhibit purely absolutely continuous spectrum on some interval. 3.4.1 A Criterion for Absolutely Continuous Spectrum Let µ be a finite measure on R; its Stieltjes (or Borel) transform F is given by F(z) = ∫ dµ(t) t − z for z = x + i y with y > 0. The following criterion has been proven in [8] for lim sup and we reproduce it here for lim inf. Proposition 3.8. Let (a, b) be a finite interval and let p > 0. Suppose lim inf y→0 ∫ b a |F(x + iy)|1+pdx < ∞ . Then µ is absolutely continuous with respect to the Lebesgue measure on (a, b). Proof. Since lim inf y→0 ∫ b a |F(x + iy)|1+pdx < ∞, there exists a sequence yn → 0 such that sup n ∫ b a |F(x + iyn)|1+pdx < C, where C is some constant. Define dµyn(x) = pi−1Im(F(x + iyn))dx. Then by [7], dµyn → dµ weakly, as n→ ∞. That is, for f a continuous function of compact support we have lim n→∞ ∫ f (x)dµyn(x) = ∫ f (x)dµ(x). Let f be a continuous function supported on (a, b), then∣∣∣∣∣∣ ∫ b a f (x)dµ(x) ∣∣∣∣∣∣ = limn→∞ ∣∣∣∣∣∣ ∫ b a f (x)dµyn(x) ∣∣∣∣∣∣ = lim n→∞ pi −1 ∣∣∣∣∣∣ ∫ b a f (x)Im(F(x + iyn))dx ∣∣∣∣∣∣ ≤ lim n→∞ ( || f ||1+1/p||Im(F(x + iyn)||1+p ) ≤ C|| f ||1+1/p . This implies that dµ(x) = g(x)dx for some g ∈ L1+p.  48 3.4.2 Bounds on the Green Function at an Arbitrary Site The following lemma proves that assuming we have a bound for the forward Green func- tions Gx(λ) for all x ∈ Tp, we can obtain a bound for all the diagonal matrix elements Gx(λ), x ∈ T, of the Green function. Lemma 3.9. Let F be the open interior of the absolutely continuous spectrum of ∆. Suppose that for any x ∈ Tp sup λ∈R(E,) E ( w1+p (〈 δx, (Hx − λ)−1δx 〉)) < ∞ , for some closed subinterval E ⊂ F,  > 0 and 0 < p < 1. Then, for every x ∈ T, we also have sup λ∈R(E,) E ( w1+p (〈 δx, (H − λ)−1δx 〉)) < ∞ . Proof. Suppose we pick an arbitrary node x0 in T and we consider its corresponding diago- nal matrix element of the Green function for the whole tree T, Gx0(λ) = 〈 δx0 , (H − λ)−1δx0 〉 . We rearrange the nodes, if needed, such that x0 becomes the origin of the tree. For this ori- gin, we have Gx0(λ) = G x0(λ). Looking at the vertices in the future of x0, we can see that after a finite number of steps, on each branch,the future tree will be a copy of the original tree. Let us denote by xi the nodes where such a copy starts. An example of such a rear- rangement is illustrated in the picture below. We know from the hypothesis that sup λ∈R(E,) E ( w1+p (〈 δxi , (H xi − λ)−1δxi 〉)) < ∞ . Starting with these nodes and using the recurrence formula for the forward Green func- tion we can work our way back to the origin, and show that the inequality holds at each intermediate node between an xi and x0. 49 x0 x3 x1 x2 x1 x2 x3 x0 Figure 3.3: Rearrangement of a tree. Let y be such a node, forward of x0 and before xi. Let ρ j be the probability distribu- tion of Gy j(λ), where y j is a neighbor of y in its forward direction. We assume inductively that E ( w1+p (〈 δy j , (H y j − λ)−1δy j 〉)) = ∫ H w1+p(z j) dρ j(z j) < ∞ . The functions that define the recurrence formula for the forward Green functions are fractional linear transformations and depend on the connectivity number of the node where the forward Green function is computed. 1. Assume y ∈ Ta ∪ {o} with ρ′ the probability distribution of Gy(λ) andK a compact set in H such that zλ is in the interior of K : E ( w1+p ( Gy(λ) )) = ∫ H w1+p(z) dρ′(z) = ∫ H×R w1+p ( −1 z1 + λ − q + 1 ) dρ1(z1) dν(q) = 1 2 ∫ Kc (∫ R w1+p (−1/(z1 + λ − q + 1)) w1+p(z1) dν(q) ) × w1+p(z1) dρ1(z1) + C . The quantity µ = ∫ R w1+p (−1/(z1 + λ − q + 1)) w1+p(z1) dν(q) does not need to be less than 1, but only bounded outside the compact set K . Using the inequalities from the proof of Lemma 3.5, µ ≤ ∫ R  Im(z1)Im(z1) + Im(zλ) ( 1 + |zλ|2 ) ( 1 + C ( 1 + |z1|2 ) ( 1 + |kq|2 )) |z1 − zλ|2  1+p dν(q) 50 which is bounded on Kc. We can therefore conclude, E ( w1+p ( Gy(λ) )) ≤ C′ ∫ H w1+p(z1) dρ1(z1) + C = C′ E ( w1+p ( Gy j(λ) )) + C . Hence sup λ∈R(E,) E ( w1+p (Gy(λ)) ) < ∞. 2. Assume y ∈ Tp with ρ′′ the probability distribution of Gy(λ) and K a compact set in H2 such that (zλ, zλ) is in the interior of K : E ( w1+p ( Gy(λ) )) = ∫ H w1+p(z) dρ′′(z) = ∫ H2×R w1+p ( −1 z1 + z2 + λ − kq ) dρ1(z1) dρ2(z2) dν(q) = 1 2 ∫ Kc (∫ R 2 w1+p (−1/(z1 + z2 + λ − q)) w1+p(z1) + w1+p(z2) dν(q) ) × (w1+p(z1)+ + w1+p(z2) ) dρ1(z1) dρ2(z2) + C ≤ C′′ (∫ H w1+p(z1) dρ1(z1) + ∫ H w1+p(z2) dρ2(z2) ) + C ≤ C′′ ( E ( w1+p ( Gy1(λ) )) + E ( w1+p ( Gy2(λ) ))) + C . Hence sup λ∈R(E,) E ( w1+p (Gy(λ)) ) < ∞. The q integral, outside the compact set K , is bounded by arguments similar to the ones in the proof of Lemma 3. When we reach the origin x0, we know the inequality holds at all other nodes. The recurrence relation for the origin is slightly different than everywhere else, due to our definition of the Laplacian. The argument that proves this final step is nevertheless almost identical to the one above.  3.5 On a recursion relation At the beginning of Section 3 we introduced quantities Ai, Bi and Ci. For q ≡ 0, they all are recursions of the following form R0(z) = 1 R1(z) = 1 + λ + z Rn+1(z) = (1 + λ)Rn(z) − Rn−1(z) 51 or, in a matrix formRn+1(z)Rn(z)  = 1 + λ −11 0   Rn(z)Rn−1(z)  = 1 + λ −11 0 n R1(z)1  . We can observe that Rn(z) has the following general form, depending on λ, Rn(z) = (Pol. of degree (n − 1) in λ) ·z+ (Pol. of degree n in λ). For λ , −3, 1 we have the following diagonal form Rn+1(z)Rn(z)  = 1det  1 1 µ2 µ1  µn1 00 µn2   µ1 −1−µ2 1  R1(z)1  where µ1,2 = 1 + λ 2 ± √( 1 + λ 2 )2 − 1 and det = 2 √( 1 + λ 2 )2 − 1. The general formula for Rn is Rn(z) = 1 det (( µn1 − µn2 ) (1 + λ + z) − ( µn−11 − µn−12 )) . Also, for n > m we have Rn(z) = µn1 − µn2 µm1 − µm2 Rm(z) + 1 det (( µm−11 − µm−12 ) µn1 − µn2 µm1 − µm2 − ( µn−11 − µn−12 )) , = µn1 − µn2 µm1 − µm2 Rm(z) + 1 det (−1)m (µ1 − µ2) ( µn−m1 − µn−m2 ) µm1 − µm2 . Lemma 3.10. The set of λ values for which either of the following identities is true is finite: (i) Rn(z) = Rm(z) = 0; (ii) Rn(zλ) Rm(zλ) ∈ R, where zλ ∈ H is the fixed point introduced in the Outline of the Proof; (iii) Rn(zλ) Rm(zλ) = −r i, where r ∈ R. Proof. (i ) Let us assume Rm(z) = 0. Since we know Rn(z) = µn1 − µn2 µm1 − µm2 Rm(z) + 1 det (−1)m (µ1 − µ2) ( µn−m1 − µn−m2 ) µm1 − µm2 , (3.15) Rn(z) = 0 iff 1 det (−1)m (µ1 − µ2) ( µn−m1 − µn−m2 ) µm1 − µm2 = 0. This identity is equivalent to µn−m−11 + µ n−m−2 1 µ2 + . . . + µ1µ n−m−2 2 + µ n−m−1 2 = µ m−1 1 + µ m−2 1 µ2 + . . . + µ1µ m−2 2 + µ m−1 2 , 52 which is a polynomial of degree max{n − m − 1,m − 1} in λ. (ii ) Using (3.15) we can write Rn(zλ) Rm(zλ) = µn1 − µn2 µm1 − µm2 + 1 det (−1)m (µ1 − µ2) ( µn−m1 − µn−m2 )( µm1 − µm2 ) Rm(zλ) . (3.16) The first term on the right hand side is a real number and since Rm(zλ) < R, the only way to obtain the desired conclusion is iff µn−m1 − µn−m2 = 0 which is equivalent to finding the roots of a polynomial of degree n − m − 1 in λ. (iii ) Relation (3.16) becomes Rn(zλ) Rm(zλ) = µn1 − µn2 µm1 − µm2 + 1 det (−1)m (µ1 − µ2) ( µn−m1 − µn−m2 ) Rm(zλ)( µm1 − µm2 ) |Rm(zλ)|2 . For condition (iii ) to be true we need µn1 − µn2 µm1 − µm2 + 1 det (−1)m (µ1 − µ2) ( µn−m1 − µn−m2 ) Re(Rm(zλ))( µm1 − µm2 ) |Rm(zλ)|2 = 0, which is equivalent to (µn1 − µn2)|Rm(zλ)|2 + (−1)m ( µn−m1 − µn−m2 ) Re(Rm(zλ)) = 0, The condition resumes to finding the zeros of a polynomial in λ.  53 Bibliography [1] M. Aizenman, R. Sims and S. Warzel. Stability of the Absolutely Continuous Spec- trum of Random Schrödinger Operators on Tree Graphs. Prob. Theor. Rel. Fields, (136):363-394, 2006. [3] R. Froese, D. Hasler and W. Spitzer. Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs. J. Func. Anal., (230):184-221, 2006. [4] R. Froese, D. Hasler and W. Spitzer. Absolutely Continuous Spectrum for the Ander- son Model on a Tree: A Geometric Proof of Klein’s Theorem. Commun. Math. Phys., (269):239-257, 2007. [5] R. Froese, D. Hasler and W. Spitzer. Absolutely continuous spectrum for a ran- dom potential on a tree with strong transverse correlations and large weighted loops. arXiv:0809.4197 [math-ph], to appear in Rev. Math. Phys., 2009. [5] F. Halasan. Absolutely Continuous Spectrum for the Anderson Model on a Cayley Tree. to be submitted, 2009. [6] A. Klein. Extended States in the Anderson Model on the Bethe Lattice. Advances in Math., (133):163-184, 1998. [7] B. Simon. Spectral analysis of rank one perturbations and applications. Mathemat- ical quantum theory. II. Schrödinger operators (Vancouver, BC, 1993) C.R.M. Proc. Lecture Notes, Amer. Math. Soc., Providence, RI, (8):109-149, 1995. [8] B. Simon. Lp norms of the Borel transform and the decomposition of measures. Pro- ceedings of the American Mathematical Society, 123(12):3749-3755, Dec. 1995. 54 Chapter 4 Conclusions Extended states, existence of absolutely continuous spectrum, was proved for Random Schrödinger operators on tree graphs. The first result was obtained by A. Klein, [6]; he proved that for weak disorder, on the Bethe lattice, there exists absolutely continuous spec- trum for almost all potentials. More recently, Aizenman, Sims and Warzel proved similar results for the Bethe lattice and quantum graphs using a different method, [1] and [2]. Their method establishes the persistence of absolutely continuous spectrum under weak disorder (and also, for the Bethe lattice, in the presence of a periodic background potential). During the same time, Froese, Hasler and Spitzer introduced a geometric method for proving the existence of absolutely continuous spectrum on graphs, [3]. They used this approach to prove the existence of absolutely continuous spectrum for the Bethe lattice of degree 3 and for a tree with strong transverse correlations and large weighted loops, [4] and [5]. The results presented in this work prove the existence of purely absolutely contin- uous spectrum for the Anderson model on different types of trees. The main ideas behind our method are based on hyperbolic geometry and were first introduced by R. Froese, D. Hasler and W. Spitzer in [3]. Chapter 2 proves extended states for the Bethe lattice of any degree K. We general- ized the result in [4] using a simplified version of the method. In [4] the quantity Z, a sum of forward Green functions is estimated, whereas we look directly at Gx(λ), the forward Green function at a vertex x. This, plus some manipulation of the expressions makes the proof of the desired estimates shorter and easy to generalize. The main advantage of out method in comparison to [4] is that we do not need a full analysis of the quantity µ2, defined in the introduction. Instead, we only need to look at the boundary points needed in the following estimates. Chapter 3 deals with a more general tree where some of the symmetry is broken. The lack of symmetry changes the analysis, making it possible to eliminate one of the steps in the proof for the Cayley tree. The tree analysed, T, has its principal nodes of degree 3 but the proof ca be extended to any degree K. Since the main goal of all this analysis on trees is to ultimately be able to give an 55 answer to the open problem of extended states for the Anderson Model on Zd, d > 2, the first step is to extend the method to graphs with loops. Our current project is extending the method to stacked trees. A stacked tree is defined as follows, we take a binary tree and an identical copy of it. We connect by an edge each vertex from the tree with its correspondent vertex on the copy and thus obtaining the stacked tree. Figure 4.1: The stacked tree Even though the loops in this graph are of a simpler nature than the ones encoun- tered on Zd, it is still a very important step towards solving the main open problem. 56 Bibliography [1] M. Aizenman, R. Sims and S. Warzel. Stability of the Absolutely Continuous Spec- trum of Random Schrödinger Operators on Tree Graphs. Prob. Theor. Rel. Fields, (136):363-394, 2006. [2] M. Aizenman, R. Sims and S. Warzel. Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder. Comm. Math. Phys., (264) 371-389, 2006. [3] R. Froese, D. Hasler and W. Spitzer. Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs. J. Func. Anal., (230):184-221, 2006. [4] R. Froese and D. Hasler and W. Spitzer. Absolutely Continuous Spectrum for the Anderson Model on a Tree: A Geometric Proof of Klein’s Theorem. Commun. Math. Phys., (269):239-257, 2007. [5] R. Froese, D. Hasler and W. Spitzer. Absolutely continuous spectrum for a ran- dom potential on a tree with strong transverse correlations and large weighted loops. arXiv:0809.4197 [math-ph], to appear in Rev. Math. Phys., 2009. [6] A. Klein. Extended States in the Anderson Model on the Bethe Lattice. Advances in Math., (133):163-184, 1998. 57

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