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Asymptotic completeness via Mourre theory of a Schrödinger operator on a binary tree grap Allard, Christine Shirley 1997

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ASYMPTOTIC COMPLETENESS VIA MOURRE THEORY FOR A SCHRODINGER OPERATOR ON A BINARY TREE GRAPH by CHRISTINE SHIRLEY ALLARD B.Sc. (Mathematiques-Sciences) Universite d'Ottawa, 1993 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF MASTER OF SCIENCE in T H E FACULTY OF GRADUATE STUDIES, Department of Mathematics We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH COLUMBIA April 1997 © Christine Shirley Allard, 1997  In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of Mathematics The University of British Columbia Vancouver, Canada Date  Aph'( 3 ^  tA /  191*  Abstract This thesis is divided as follows: The first chapter introduces the main ideas intuitively while assuming an acquaintance with quantum mechanics. The second chapter exposes the mathematical setting of the problem under investigation and contains a brief excursion into selfadjointness of unbounded operators. The core of the thesis is contained in the third chapter where a conjugate operator A is defined in order that the Mourre estimate for the discrete Hamiltonian H = L + Q is shown to hold for a binary tree configuration space, where L is the discrete Laplacian and Q a discrete potential either of short-range type or of long-range type satisfying a first order difference condition. In the last chapter the result from chapter three as well as some extra material is used to show that asymptotic completeness holds for a onebody system having a binary tree configuration space with either of the following three types of potential: 1) long-range potentials satisfying first and second order difference conditions 2) short-range potentials of order o(|w| ) and satisfying a second order difference condition or 3) short-range potentials of order o(|u|~ ). -1  2  ii  Table of Contents Abstract  ii  Table of Contents  iii  Acknowledgment  v  Chapter 1.  1  Introduction  Chapter 2. Definition of H and Self-adjointness 2.1 Initial definitions 2.2 Self-adjointness 2.3 Unbounded operators  4 4 5 7  Chapter 3. The Mourre Estimate 3.1 Spectrum 3.2 Definition of the Mourre Estimate 3.3 The Mourre Estimate for L 3.3.1 Unitary equivalence 3.3.2 Definitions of A and A and the Mourre Estimate for the line Laplacian . . . 3.3.3 The Mourre Estimate for L acting in Z (Z+) 3.4 Decomposition of 1 {V) 3.4.1 Action of L on © l {S ) 3.4.2 Bases for l (S ), n = 0,1, 3.4.3 Invariant subspaces Mk 3.4.4 Definition of the conjugate operator A 3.4.5 The Mourre estimate for L acting on l (V) 3.5 Mourre Estimate for H acting on l (V) 3.5.1 Rate of decrease of the "differences" of Q on 1 {V) 3.5.2 Rate of decrease of Q itself  9 9 11 13 13 14 15 17 18 19 21 24 26 27 30 32  Chapter 4. Asymptotic Completeness 4.1 Asymptotic Completeness 4.1.1 Construction of wave operators 4.2 The limiting absorption principle 4.2.1 Added fourth condition 4.2.2 Classical and less classical approaches to the L A P 4.3 i?-smoothness  34 34 35 37 37 38 39  Bibliography  41  2  2  2  n  2  n  2  2  2  Appendix A.  Derivation of A  43  iii  Table of Contents Appendix B. The double commutator [[Q, iA], iA] B . l Definitions B.2 Derivation of the double commutator  iv  50 50 50  Acknowledgment I would like to thank my mother, my sister, my brother and my dog Wolfie for their unconditional support through dark times, in particular my mother. Further I would like to thank Stan Page for his encouragement and attention and John Fournier for his patience. I would also like to thank Michael Kennett who got me started on Latex and Anders Svenssonforhis expert advice on the intricacies of Latex. Finally I would like to thank the second reader for an in depth reading of my thesis as well as my supervisor for his financial support throughout.  Chapter 1 Introduction  If two initially free quantum particles collide at time t = 0, through an interaction potential which approaches zero for large separation, then these particles are expected to either bind together to form a bound state or depart after interaction to resume their course as free particles. Roughly speaking the property that colliding particles or systems of particles resume their course as free particles or systems of particles after interaction is called asymptotic completeness. This is one of the three main problems of rigorous scattering theory. The two others are that of existence of wave operators and absence of singular continuous spectrum. For the system of two quantum particles obeying Schrodinger's equation, the existence problem was solved by Cook and is known as Cook's method or a variant of it. The problem of absence of singular continuous spectrum has had a somewhat more sinuous history. It is Lavine in [L] that had the idea of using commutator methods to obtain absolutely continuous spectrum. Mourre [Mol-2] further had the idea of localizing in energy the commutator method to obtain more succinct results. As for asymptotic completeness, again Lavine [L] used a combination of the commutator method and the theory of smooth operators to obtain completeness of the wave operators for a certain class of potentials. In addition to earlier stationary methods, many different time-dependent methods [En] emerged in order to obtain asymptotic completeness for one-body, two and three body and finally for the N-body problem for a certain range of potentials. We refer the reader to [Si] for an in depth discussion of the latter. We are concerned in this thesis with the one-body problem for a discrete Schrodinger operator H with a binary tree configuration space as shown infigure1 below, that is to say, a  1  Chapter 1. Introduction  system in which an electron placed in an external potentialfieldmoves from site to site on an infinite binary tree. z  Figure 1  This binary tree configuration space is called either an infinite "regular" Cayley tree or a Bethe lattice. The number of branches from an arbitrary vertex v to the origin of the tree is important in this work and is taken to be the distance of this vertex to the origin denoted by |v|. The reason for considering such a lattice is related to what is called the Anderson approximation problem. Anderson examined a renormalized perturbation series in order to study the stability of localized states. It turns out that the first or second order approximations for a real lattice are exact for the Bethe lattice [AAT]. We start in chapter two by setting up the problem mathematically then proceed in chapter three in showing that the Mourre estimate holds for H under certain conditions on the potential. We prove the following theorem: Theorem 1 If H = L + Q where L is the free discrete Laplacian operator and Q a discrete potential of the form Q\ + Q2 where both Q\ and Q2 are multiplication operators such that  \v\q\(v) —»• 0 as \v\ —> 00 and such that q2(v) —> 0 and \v\\q2(v) — q2{w)\ —)• 0 as \v\ —» w any vertex in either the preceding or successive level, then the Mourre Estimate for H as expressed in section 3.2 holds o .  A word needs to be said about the method used. The Mourre estimate amounts to the definition of a conjugate operator A that does the trick. When working on a binary tree graph, one can either proceed via a decomposition of the underlying Hilbert space into levels of constant \v\ in order to work with direct sum formulations of the operators under investigation 2  Chapter 1.  Introduction  (including the conjugate operator) or take a more direct approach via a discrete formulation of the operator A. In the former case, the definition of the conjugate operator is given by: A = i(IT*(2(X - N) + 1) - [2(X -N) + 1)11) = i(A* - A), where EE and IT are level raising and lowering operators, X is a multiplication operator by the distance to the origin and N is an operator related to the decomposition at hand. Both approaches are equivalent and are used when convenient. In the last chapter we proceed in obtaining asymptotic completeness for H with longrange potentials satisfying first and second order difference conditions or short-range potentials either of order o(|v| ) and satisfying a second order difference condition or of order o(|u| ). _1  -2  We prove the following theorem:  Theorem 2 Let H = L + Q, where Q is of the form Qi + Q2 + Qs where Q\ and Q2 are as in theorem 1 and Q 3 a multiplication operator such that qz{v) —>• 0 as \v\ —> 00 and \2qs{w) — QsM — 93(-z)I decays faster than -r-To  for w as in theorem 1 and z in either of the preceding  or successive second levels, then asymptotic completeness holds for H. In other words the onebody system for a discrete Schrodinger operator under the above conditions on a binary tree configuration space is asymptotically complete o  Finally, we would like to point out some interesting conditions on the potential that arose in the course of this thesis. It so happens that although the condition that |v||g2(w) — 92 (w) I —> 0 as \v\ —> 00 refers to all neighbors w of v, not all neighbors need satisfy this condition. This leaves rather seemingly weak conditions on the part of the potential. One may ask whether this observation can lead to more precise results about the spectrum of random Schrodinger operators on a Bethe lattice? As we leave this thesis we embark on this new journey of discovery.  3  Chapter 2 Definition of H and Self-adjointness  As was stated in the introduction, we will work in this thesis with a discrete Schrodinger operator H, having a binary tree configuration space. In this chapter we introduce the notions necessary for a proper definition of H. Further, we will proceed in showing that our Schrodinger operator is self-adjoint. Recall here that self-adjointness of H allows a unique solution to the Schrodinger equation with initial condition and hence is a crucial property. Further, according to von Neumann's theory of quantum mechanics, self-adjoint operators are the basis of observables, which allow for a physical interpretation of the spectrum. Hence this is the starting point: a self-adjoint operator to work with.  2.1  Initial definitions  The first two definitions are those of the configuration space consisting of the graph G and of the Hilbert space "H on which H acts. • Let G — (V,E) be the infinite binary tree composed of vertices V = {v} and edges E = {e}. • Let l (V) be the Hilbert space consisting of all C-valued square-summable sequences 2  4> = (<f){v)),v € V, i.e. all sequences such that Y^, \4>{ )\ < v  2  0 0  v  Next we introduce the free discrete Laplacian and its perturbation, the potential operator. • Let L be the Laplacian of the graph G = (V, E) given by {Lcj>){v) = w\a =l  - WW  o r  vw  4  =  - ( ))M ) D  G  V  I-) 2  1  Chapter 2. Definition of H and Self-adjointness where A(G) and D(G) are the adjacency and degree matrices corresponding to G. Recall 1 if there is an edge connecting v and w ^ 0 if there is no edge connecting v and w and d(v) is the degree of v, i.e. the number of edges surrounding v. To be more precise, let {e (w) = 5 } be a complete orthonormal system in l (V). And 2  v  WV  define Le (w) = a v  where d  wv  — Oiiw^v  wv  - d  wG V  wv  and d(v) ifw = v. Then L can be extended by linearity to L on e  the dense subspace D(L ) of I spanned by the e 's. 2  e  v  Furthermore L is symmetric thus e  closable. Consider its closure L to be the operator we are working with i.e. L = L . e  e  • Let Q be the potential operator acting on 1 {V) given by a real multiplication operator, 2  so (Qu)(v) = q(v)u(v). Finally, with this at hand we are ready to state our main object of study : • Let H = — L + Q be the Schrodinger operator acting on l (V) with L and Q as above. 2  At this point no conditions have been imposed on Q. As was stated in the introduction, these conditions on Q play a fundamental role in the properties that can be attributed to our operator. We assume in this thesis that the potential is either of long-range type satisfying first and second order difference conditions or of short-range type either of order o(|u| ) and _1  satisfying a second order difference condition or of order o(|v|~ ). Recall that short-range refers 2  to the rate of decay at infinity of Q being faster than py, where \v\ denotes the distance from the vertex v to the origin, specifically the number of branches from the vertex v to the origin and long-range refers to a rate of decay slower than yj.  2.2  Self-adjointness Having defined the Schrodinger operator we are working with, we now ask, what con-  ditions should be imposed so that it be self-adjoint, i.e. when is H = —L + Q self-adjoint? 5  Chapter 2. Definition of H and Self-adjointness Self-adjointness is important for in this case the eigenvalues of the operator are actually real values which yield a possible straightforward physical interpretation, as mentioned earlier. Let's start with self-adjointness of the free Laplacian operator L. Prom earlier considerations we had that {L4>){v) = {{A{G)-D{G))<t>){v) acting on vectors in l (V) by matrix multiplication. One can obtain a straightforward result on 2  self-adjointness by using the following theorem by Schur in the case the degree of the vertices are uniformly bounded i.e. d(v) < M V v Schur's Theorem Let (a,ij) be an infinite matrix of scalars such that oo  (a)  ^2\oij\<Mi,  * = 1,2,...  and oo  (b) ^ M < M ,  j = l,2,...  2  Then the matrix represents an operator A on l (V) to 1 {V) such that 2  \\A\\<  2  (MiAf )5. 2  In the case of the binary tree, we have that d(v) < 3 W, so in considering the infinite matrix (l ) — (a vw  vw  — d ) for w € V given above we see that both (a) and (b) are satisfied vw  by taking Mi = M2 = 6. Thus, the operator L represented by this matrix is bounded and ||L|| < 6. Working in the free case with a bounded operator makes things easier than working with unbounded operators. Since L is bounded and symmetric and acts on the whole of 1 {V) 2  then L is trivially self-adjoint. Next look at the Schrodinger operator H = — L + Q. When is H self-adjoint? More precisely what conditions on Q should be imposed such that H is self-adjoint? Since Q is symmetric, using Schur's test once again, wefindthat (a) and (b) are satisfied under the extra hypothesis that Q be bounded, i.e. that \q(v)\ < K for all v and for some constant K. Hence 6  Chapter 2. Definition of H and Self-adjointness being bounded and symmetric, H is self-adjoint. Having looked at the bounded case we proceed to the unbounded case.  2.3  Unbounded operators Recall that an unbounded operator is discontinuous at every point in its domain. When  is such an operator self-adjoint? A useful criterion for self-adjointness is based on the so called deficiency indices. First recall that in the bounded case M was self-adjoint if for all f and g  € H we had (2.2)  (Mf,g) = (f,M*g) and M = M*.  Also since both operators are defined on all of Ti, D(M)  = D(M*).  In the  unbounded case we would like to use the same defining relation (2.2) for all / € D(M) and g € D(M*). Recall that D(M*) is given by all elements g G H such that there exists a vector g* EH with  (Mf,g) = (f,g*) for all / G D(M)  (2.3)  In this case we define g* = M*g. Now what do we need for M * to be well-defined ? Suppose there exists g* and g\ such that (2.3) holds and hence (/, g\ — g%) — 0 for any f Eli.  So our  question becomes what do we need to obtain g\ = g\ or h = g\ — g\ = 0 ? For this to happen, it is sufficient that D(M) be dense in Ti. For in this case, the orthogonal complement to D(M) is zero. Hence h = 0 and g{ = g^. So density of D(M) is implicitly assumed whenever adjoints are dealt with. As for self-adjointness, an operator is self-adjoint if M = M*. Suppose we are dealing now with a densely defined symmetric unbounded operator M from a Hilbert space into itself.  Recall that a densely defined operator L is symmetric if  (Lf, g) = (/, Lg) for all / and g in D(L). Further recall that in the case of a bounded operator, with domain and range equal to Ti symmetry is equivalent to self-adjointness whereas for an unbounded operator this equivalence does not necessarily hold, as it may happen that D(M)  7^ D(M*).  For example the operator associated with i-jj^ defined on the set {/  €  L [0,27r] I / is absolutely continuous on [0, 2TT] and /(0) = /(27r) = 0} is symmetric but not 2  7  Chapter 2. Definition of H and Self-adjointness self-adjoint; see [AJS] for details. Hence we see that defining the domain of an unbounded operator is no little thing and that the restriction D{M) = D(M*) is severe in the case of unbounded operators! The operator M itself, may or may not be selfadjoint. In the case it is we are done. In the case it isn't, (that is to say D(M) ^ D(M*) as in the example above) since D(M) is a proper subset of Ti we may ask ourselves if it is possible to extend the domain of definition of M (i.e. D(M)) so that this extension of M on this new subset of Ii be selfadjoint ? The answer as will be seen depends on the operator. It may happen that no self-adjoint extension exists regardless of how we try to extend D(M), it may happen that there are many different selfadjoint extensions or it may happen that there is a unique selfadjoint extension, in which case the operator is called essentially self-adjoint. Question : How to pin down in a nutshell when an unbounded symmetric operator has selfadjoint extensions ? The answer lies in a pair (m, n) of numbers associated with the operator called the deficiency indices. As the word suggests these numbers are a measure of how deficient the operator is in being selfadjoint. Definition The deficiency indices (m, n) associated with the operator M are given by m = codim Tl(M + i) K  n  — codim TZ(M — i)  (2.4)  To see why the word deficiency is used, recall briefly that for a symmetric and closed operator M , U = K{M - i) ®N{M* +%) = n{M + i)Q N{M* - i). So that in fact m = dim M{M* - i) and n = dim N(M* + i). These spaces are called the deficiency spaces of M. Further recall that if M is selfadjoint 1Z(M ± i) = H. Thus we see that the values of m and n act minimally as indicators by which it is possible to evaluate the extent to which M itself is selfadjoint. That is if (m, n) = (0,0) then M is essentially selfadjoint. Further M has self-adjoint extensions iff m = n. We refer the reader to [HP]. We will use this nice theory in the next chapter so as to show that a specific operator we deal with is indeed selfadjoint.  8  Chapter 3 The Mourre Estimate  One of the very crucial and interesting subjects in the theory of operators is the determination of the spectrum of a given operator. This chapter is devoted to a mathematical result which provides explicit information on the spectrum of a self-adjoint operator as well as a means to obtain asymptotic completeness: the Mourre estimate. Let's step back a bit and expose a brief profile of the spectrum of a self-adjoint operator, before going any further.  3.1  Spectrum Recall that the spectrum of a self-adjoint operator is real and can be decomposed in sev-  eral manners, as follows: firstly recall that a spectral family {E\} for X in the spectrum of H, can be associated to every self-adjoint operator, via the spectral theorem. This spectral family determines a spectral measure which for fixed <f> G Ti can be used to construct a non-negative countably additive Borel measure, dp,^ = dfi [Ro], But who says measure says decomposition. According to the Lebesgue decomposition theorem, this measure can be uniquely decomposed into respectively an absolutely continuous and singular part with respect to the Lebesgue measure on R, d/i = dfi + dfj, . Further, any Borel measure can also be decomposed uniquely ac  s  into a pure-point part and a continuous part. This leads to the decompositions of % into T-L ffi% = H ac  s  pp  © T-L . Further since H c  ac  C H and since we wish to distinguish the different c  parts of H , we arrive at the following decomposition: c  H = Hac © Use © Hpp  9  Chapter 3. The Mourre  where H  Estimate  = "H 0 Ti  sc  c  a n d where 4> £ %pp> 'rise a n d Hac if d / V * respectively pure-point, s  ac  singular continuous or absolutely continuous. T h u s i n spectral terms, where a specific type of s p e c t r u m is defined t o b e a(H\-n ) x  for x either pure-point, singularly continuous, absolutely  continuous, continuous, or singular, we get that a(H) = a (H) ac  U o-sc(H) U o-pp(H), where o (H) ac  i.e. c r ( i ? | ^ ) , a (H) a c  (H\Hac<8H )i  a  sc  pp{H)  eigenvalues a n d a  sc  c  U o- (H)  = a (H)  pp  ac  U  o (H) s  denotes the absolutely continuous s p e c t r u m of H,  the singular s p e c t r u m i.e.  s  a  = o (H)  a (H)  the point spectrum, i.e. a  c  pp  = a(H\ ) npp  the continuous s p e c t r u m i.e. = o~p~ where a  v  the singular continuous spectrum given by  is the set of  a (H)\a (H). c  ac  H a v i n g this i n m i n d suppose now that the operator we are dealing w i t h is a "generic" Schrodinger operator H = L + V, where L is either the discrete or continuous L a p l a c i a n a n d V is a p o t e n t i a l operator. U n d e r specific conditions o n V, H can be shown to be selfadjoint. W h a t c a n be s a i d about its spectrum? It w i l l depend o n how the p e r t u r b a t i o n t e r m behaves. R e c a l l that i n the discrete case of a Schrodinger operator acting o n a line we h a d (Hu)  n  If q = 0 t h e n H is absolutely continuous, i.e. H = % n  a c  = (Lu)  n  + q. n  . A n d i f q —>• 0 one might expect t h a t n  H s t i l l has absolutely continuous s p e c t r u m as well as some eigenvalues. However, B . S i m o n i n [S] a n d P o s c h e l i n [P6] have given examples of respectively power decaying potentials, \q \ = n  0(n / - 1  2 + e  ) for any e > 0 a n d l i m i t periodic potentials given by a general c o n s t r u c t i o n , such t h a t  H (a slight variant i n [P6]) has pure-point spectrum. T h e question of whether a specific operator has continuous s p e c t r u m a n d of what type (absolutely continuous or singularly continuous), is i m p o r t a n t i n scattering theory, for the scattered states are represented by state vectors l y i n g i n the absolutely continuous subspace of H. Hence existence of absolutely continuous s p e c t r u m points to existence of states other then b o u n d states, namely scattered states.  W i t h respect to the s p e c t r u m of a Schrodinger operator, the v i r i a l theorem clearly is a useful theorem i n a n a l y z i n g it.  B y use of it, absence of eigenvalues i n certain regions of  the s p e c t r u m of specific Schrodinger operators has been obtained, [We] for example. F u r t h e r M o u r r e i n [ M o l ] has given a criterion by w h i c h when combined w i t h the v i r i a l theorem, H has finite p o i n t s p e c t r u m w i t h eigenvalues of at most finite m u l t i p l i c i t y a n d empty singular continuous s p e c t r u m o n a n interval i" where the criterion holds. T h i s implies for example that  10  Chapter 3. The Mourre  Estimate  eigenvalues of H c a n not accumulate i n / , see [ C F K S ] for details. T h i s c r i t e r i o n was s i m p l i f i e d i n [PSS] a n d is know i n the literature today as the M o u r r e estimate. I n the next sections we proceed i n exposing the M o u r r e estimate a n d showing that o u r discrete Schrodinger operator of the previous chapter satisfies it.  3.2  Definition of the Mourre Estimate  Definition 3.1 The self-adjoint operator H satisfies a Mourre estimate at X if there is a selfadjoint operator A (called the "conjugate operator") such that 1. D(A) n D(H) is dense in D(H) in the norm \\(H + i) • \ \ 2. The form Rf{(i)[H,iA]RH{i)  is bounded, where -R#(J) = (H —  is the resolvent corre-  sponding to H. 3. There is an interval I containing A such that Ei(H)[H,iA]Ej(H)  > 0Ej(H)  2  + K where  9 > 0, K is a compact operator and Ej(H) is. the spectral projection associated to H on the interval I.  W e make a note here about the " m e a n i n g " of conditions one a n d two, specifically we  C l a i m Condition 1 ensures that the form Rjj(i)[H,iA]RH(i)  is well-defined, namely that the  set S = {4>: RH(i)(f> G D(A) D D(H)} is dense in Ti. Proof A s s u m e c o n d i t i o n 1 holds. T h e n V <j) € D(H) 3 <t> € D{A) n D{H) w i t h n  \\(H + i){(j)-4) )\\  0.  T h i s w i l l show that  S is dense. B u t R (i)^  n  N o w let  n  H  \\{H + i)(<p - {R„{i))^)\\ n  (j) G D(A) n D(H), soip eS n  ip <E U. W e w i s h to find a sequence i/) £ S w i t h ||V'„-'0|| ~> 0.  n  €  D{H) so 3 4> e D(A) n D{H) such that n  -> 0.  = \\{H + i)4> n  Let  {H + i)cj> = n  Then  R [i)%l> = H  and |k/>„ - ^ | | -+ 0 o  A consequence of the M o u r r e estimate w i t h a n added c o n d i t i o n o n the double c o m m u t a t o r [ [ i ? , i A ] , i A ] is the l i m i t i n g a b s o r p t i o n p r i n c i p l e , w h i c h implies absence of s i n g u l a r continuous s p e c t r u m . W e w i l l talk about this a b i t more i n the next chapter. W e w i l l see that this estimate is a very powerful t o o l i n a n a l y z i n g the s p e c t r u m of a n operator.  11  n  Chapter 3.  The Mourre  Estimate  For the two-body p r o b l e m , i n the continuous case a conjugate operator is found to be:  A = ^(X-P  +  P-X)  where X is the p o s i t i o n operator a n d P the m o m e n t u m operator given formally by P<j> = for 4> £ £>(V). M o r e specifically it is defined by  A<f> = -i^-U <l> |t=o at t  te R  where the (£/*(£) (x) = e \ (p(e x) define a strongly continuous one-parameter group of operators nt 2  t  called the group of dilations on L ( M ) . A is the generator of the group of dilations, or briefly 2  n  the generator of dilations. W h e n w o r k i n g w i t h graphs, it is n a t u r a l ask : for w h i c h graphs does a M o u r r e estimate hold? In this chapter we w i l l show that the M o u r r e E s t i m a t e holds for our discrete operator H = —L + Q acting o n G where G is a binary tree graph. In the process, we w i l l uncover the conditions to be imposed o n Q. It w i l l t u r n out that Q must be of the type Q = Q\ + Q2 where Qi is decaying faster t h a n j—y so |v|gi(v) —> 0 as \v\ —>• 00, a n d Q2 is such t h a t 92 (i>) —> 0 as |v| —>• 00 a n d \q2( ) — Q2(w)\ is decaying faster t h a n 7^7, where \v\ is the distance f r o m the vertex v  M v to the o r i g i n a n d w is i n either the preceding or successive levels, as noted i n the i n t r o d u c t i o n . T h e first step i n showing that the M o u r r e estimate holds for a "generic" Schrodinger operator H — L + Q , is showing that the M o u r r e estimate holds i n the "free" case, i.e. for L. I n the case the u n d e r l y i n g configuration space is discrete a n d slightly more c o m p l i c a t e d t h a n a regular g r i d this isn't as straightforward as i n the case where this space is R . T h e reason is n  that a conjugate operator i n these discrete cases is not as readily found.  A strategy to follow, when working w i t h a b i n a r y tree g r a p h , is to decompose the o r i g i n a l H i l b e r t space i n such a way that it be expressed as a direct s u m of subspaces o n w h i c h the L a p l a c i a n L is u n i t a r i l y equivalent to a direct s u m of "half-line" L a p l a c i a n s , C acting o n  Z (Z ). 2  +  Hence to show that a M o u r r e estimate holds for L acting on l (V), the first step (after the 2  d e c o m p o s i t i o n of l (V)) is to show that the M o u r r e estimate holds for the "half-line" L a p l a c i a n . 2  T h e next step is to show that the M o u r r e estimate holds for the direct s u m (hence for L a c t i n g 12  Chapter 3. The Mourre  Estimate  on l (V)). Andfinallyusing this fact, show that the Mourre estimate holds for the Schrodinger 2  operator H = L + Q. A conjugate operator for the graph can then be defined as a direct sum of conjugate operators on the half-lines. Further, we will see that it is also possible to obtain a more tangible definition of the conjugate operator. We will take each of these steps throughout the chapter. Let's start by the half-line case, so as to get a taste of the Mourre estimate, and see later how it arises through the decomposition of I (V). 2  3.3  The Mourre Estimate for L As a preliminary step to showing that the Mourre Estimate holds for L acting on / ( Z ) 2  +  which we will denote as C, i.e. the half-line case , we start by showing that it holds for L acting on / ( Z ) , which we denote as L, then show how the half-line case emerges from the former. 2  3.3.1  Unitary equivalence Actually we show that a Mourre Estimate holds for a unitarily equivalent operator L.  Let's recall the definition for unitary equivalence: Let B and C be two operators acting on Hilbert spaces X and Y respectively, then  Definition 3.2 C is said to be unitarily equivalent to B if there exists a unitary operator T : X -> Y (i.e. T is an isometry such that TT* = I = T*T) such that C = TBT"  1  = TBT*.  Next, recall that the the discrete Laplacian is given by : L : / ( Z ) ->• Z (Z), (L0)(n) = 0(n + l) + 0 ( n - l ) - 2 0 ( n ) 2  2  (3.1)  Let T be the unitary equivalence given by T : / ( Z ) —> L (0,2ir) where 2  (T(p)(t) =  2  X ^ - o o 4 ( )e - Let's calculate how L is transformed under this equivalence : >  n  ini  We need to calculate TL(4>(n)) for (</>(n)) € Z (Z). But after re-indexing, we have 2  W(n)))  =  ^Eit~oo^(*)(e *  -  ^2Zt=-oo^{t)-2)<t>{k)e  =  (2cos(t) - 2)4>(t).  < (  + ^*- >'-2c ')  + 1 ) t  1  ikt  13  t t  (3-2)  Chapter 3. The Mourre  Estimate  Thus L is transformed to L, the multiplication operator by 2cos(i) — 2. The importance of unitary equivalence comes from the consequence that if such a T can be found for a pair of operators B and C then B and C have the same "internal structure". So if working with C rather than B makes things more transparent then this can be done by virtue of this "equivalence". In particular, we will choose to work with either L rather than L when it is more convenient.  3.3.2  Definitions of A and A and the Mourre Estimate for the line Laplacian  A n initial guess to what A , corresponding to L , should be is:  A= -{XV  + VX)  l  (3.3)  where A acts on l (Z), X4>(n) = n(j)(n) and T>4>(n) = 4>(n + 1) — 4>(n — 1). Since X and V are 2  unitarily equivalent, under T of section 3.3.1, respectively to — i-^ and —2zsin(i), we readily obtain the unitarily equivalent form of A:  A = i(sin(t)-^ + |-sin(t)) We claim that A is essentially self-adjoint on C ( 5 ) , where S  (3.4)  0 0  1  1  is the unit circle associated to  the interval (0,27r). To see this, we can apply the deficiency index theory we talked about in chapter one. In particular we need to solve the differential equations A*<f> = ±i<f> for 4> G D(A*) and check whether the solutions cf> are in L (0,27r). The adjoint A* here is formally given by the 2  same formula (3.4) as above but its domain D(A) contains </>'s which are absolutely continuous away from 0 and IT. We need to solve A*(f> = \(2s'm(t)(j)' + \ cos(t)cf)) = ±i(f>. We obtain  =  -77mJ2 * L (0,2K) and </> = ^ " ^ f ^  1  2  +  £ L (0,2TT), which yield (0,0) as 2  deficiency indices. Thus this is the one!  Remark In analogy to the continuous case, another guess for the conjugate operator would be  where the differential operator  acts on L (0,2ir) 2  with say periodic boundary conditions.  Again, deficiency indices may be calculated by solving the two differential equations: 14  Chapter 3.  The Mourre  Estimate  \{t<j>' + \4>) = ±i(j). Solutions to the latter are <j> = t~% and <£_ = t^, which yield (0,1) as +  deficiency indices, as t~f ^ L (0,2n). Hence it is not possible to find a self-adjoint extension of 2  C as defined in (3.5). Another reason that (3.4) is an appropriate conjugate operator is that we also want [L,iA] to be positive. Suppose L = uj(t) is any multiplication operator, then if we let A=-  +  w  e  §  e t  * * t-^' na  = (u/) which is positive except for thresholds t where 2  u)'(t) = 0. Since L = 2cos(£) — 2 = w(t), we see that the form of A in (3.4) is an appropriate choice for a conjugate operator. Either of the representations (3.3) or (3.4) of the conjugate operator will do for the Mourre estimate. We are now ready to show that the Mourre estimate holds for L and hence for L. Start by observing that if we consider L = 2cos(£) — 2 acting on L (0, 2TT) into L (0,2ir) and A as 2  2  given above then [L, iA] = 4sin (i) as noted above and conditions 1 and 2 in the definition of 2  the Mourre estimate are satisfied. Since D(L) = W, condition 1 boils down to D(A) being dense in W. = L (0,2n), which 2  holds for A selfadjoint. Condition 2 follows from the fact that |X,iA] = 4sin (£) which is 2  bounded on L (0,2n). 2  Next let / be an interval away from the threshold t E {0,7r}. Then take 6 = min{4sin (£)}. 2  Then the third condition is satisfied with K = 0. We are now ready to witness the emergence of the half-line case from the whole-line case! Note that in calculating [ i , i A ] the term involving 4>(n), i.e. the constant diagonal term, just cancels out, hence from here on, without loss of generality, we consider L " = "4>{n + 1) + 4>(n — 1), reneging the constant degree term in the void. This makes the use of the decomposition to come more transparent.  3.3.3 The Mourre Estimate for L acting in Z (Z+) 2  Recall that to show that the Mourre estimate holds on the binary tree, the idea is to first of all reduce L acting in l (V) to a direct sum of "1-D" half-line Laplacians, C acting on 2  15  l (Z ). 2  +  Chapter 3.  The Mourre  Estimate  T h i s operator £ , turns out to be given by  C : Z ( Z + ) ->  / (Z+)  2  2  (3.6) ( a , a i , a , a , •..) 0  2  (ai,a  3  0  + "2,01  +  a ,...) 3  S o w e n e e d t h e M o u r r e e s t i m a t e t o h o l d , i n p a r t i c u l a r f o r t h i s C.  A s w e k n o w it h o l d s for  w h o l e line, the p o i n t is to get f r o m the w h o l e line to the half-line. to  find  C : 1 {Z ) —>• Z 2  a n expression of the half-line L a p l a c i a n  whole-line Laplacian L  ->  : 1 {Z) 2  Z (Z).  +  2  In particular we will  (Z ) +  i n (3.6)  want  in terms of  T h i s will give us the M o u r r e estimate for C  2  the  the  via  the  M o u r r e e s t i m a t e for L t h a t a l r e a d y holds f r o m the p r e v i o u s section. F o r c e r t a i n details, t h o u g h , we will want  t o u s e t h e u n i t a r i l y e q u i v a l e n t e x p r e s s i o n o f L,  T o o b t a i n t h e d e s i r e d e x p r e s s i o n f o r C v i a L,  L.  t h e i d e a is t o u s e t h e d e c o m p o s i t i o n o f Z ( Z )  as a  2  d i r e c t s u m o f e v e n £ a n d o d d O f u n c t i o n s u b s p a c e s as f o l l o w s : W e first r e m a r k t h a t O is i n v a r i a n t under  L.  T h i s c a n be seen in either  of two  ways:  1)  one can note that  the  corresponding  u n i t a r i l y e q u i v a l e n t s p a c e o f o d d f u n c t i o n s i n L (0,2ir) i s i n v a r i a n t u n d e r L = 2 c o s ( t ) . T h u s b y 2  u n i t a r y e q u i v a l e n c e O i s i n v a r i a n t u n d e r L. of L  i n (3.1)  e l e m e n t o f O,  Alternatively one c a n use the discrete  2)  to readily obtain the same result. t h e n L(...  T h a t is let (...  - a o , 0 , a o > a i ; ...)  , —QI,  w h i c h i s a g a i n a n e l e m e n t o f O.  T h e n , since O  =  (•••  , — a i , — a o , 0, a o , a i , • • •) b e  > ~( 2  + ao))  a  is i n v a r i a n t  O n e can verify that  —  <*i,0,ai, a  2  +  an  ao,...)  u n d e r L , w e c a n c o n j u g a t e it  t h e i n c l u s i o n U : ( a o , a i , 02, . . . ) ! - > • ( . . . , — Q i , — a o , 0 , a o , a i , . . . ) n a m e l y the one we are l o o k i n g for £ .  definition  by  to obtain a half-line operator,  C = U~ LU, l  t h a t is w e h a v e as  in  (3.6):  (£a)  It  n  = a _i n  + a  n  +  i  (a_i  =  0)  r e m a i n s t o define t h e c o n j u g a t e o p e r a t o r o n t h e h a l f line, A,  Mourre  estimate.  i-(XV + VX) 2  under  A  Recall that  the  conjugate  or under unitary equivalence by  as well,  A  operator  o n the  (3.7)  that will do the trick whole  line was given by  A = \(s'm(t)^~ + -7- s i n ( t ) ) . i dt dt  is o b t a i n e d i n a s i m i l a r f a s h i o n as £ ,  16  for  namely, by  Since  setting  O  the  A  =  is i n v a r i a n t  A = U~ AU. 1  Chapter 3.  The Mourre  Estimate  F u r t h e r , using the definition A = \(XV A:  + VX), we o b t a i n that :  (a ,ai,O!2,...) ^ 0  (|)(3ai,5a - 3 a , 7 a 2  0  3  -5ai,...)  (3.8)  that is  [(2n + 3 ) a  n + 1  - (2n + l)a _i]  (a_i = 0)  n  W e point this o u t here, b u t w i l l use this information later o n when showing that a M o u r r e estimate holds for  l (V). 2  Hence, we have what we need to get the M o u r r e estimate for the half-line L a p l a c i a n . Since we k n o w t h e M o u r r e estimate holds for L, we need only start w i t h this t h e n a p p l y U t o o b t a i n the M o u r r e estimate for C: W e use also the fact that  U'^E^VjU  = E^U^LU)  = E (C) t o T  obtain:  U~ E (L)UU~ [L,iA]UU~ Ei(L)U 1  1  >  1  I  £/(£)[£, U ] £ ( £ ) 7  where  0 is unchanged a n d U~ [L,iA]U l  = [U~ LU,iU~ AU] l  l  0lJ- E {L)U l  2  6E {C)  (3.9)  2  >  = [C,iA].  Note Since the conjugation m a n i p u l a t i o n entails no unboundness, conditions 1 a n d 2 o f the M o u r r e estimate are satisfied. Hence we see that i f the M o u r r e estimate holds o n the whole line, it also holds o n the half-line. W e now proceed t o expose to daylight the decomposition of 1 {V) that we mentioned 2  earlier.  3.4  Decomposition of l (V) 2  R e c a l l once again! (you are b o u n d to know this by now) that when w o r k i n g o n the tree, t h e idea is t o decompose the H i l b e r t space 1 {V) where V is the set of vertices o f the tree, i n such 2  a way that (the off-diagonal part of) the discrete L a p l a c i a n L a c t i n g o n l (V) given b y 2  (3.10) be u n i t a r i l y equivalent b y means of invariant subspaces o f l (V) 2  t o a direct s u m of half-line  L a p l a c i a n operators arising as restrictions of L t o these subspaces. Hence the forgoing analysis  17  Chapter 3. The Mourre  Estimate  can be applied. Furthermore this also gives a way to study the full spectrum of the Laplacian, namely by studying the spectra of the restrictions of L to each of these invariant subspaces. We start by decomposing the space 1 {V) in a direct sum of Hilbert spaces l (S ) where the 2  2  n  5„'s represent "spheres" or levels arising from a vertical splintering of the tree i.e. S  = {v : \v\ = n } , where \v\ is the distance from the origin to v. Therefore l (V) = 2  n  ® ~ / ( 5 ) . See figure 2 below. 2  = 0  n  Figure 2  Let's discuss the structure of these "spheres" a bit more. Each l (S ) is a finite-dimensional 2  n  Hilbert space which is spanned by 2 orthonormal basis elements. One example of such a n  basis is the standard delta basis. For the sake of invariance though, we will want to choose an alternate basis; the discrete Haar basis. Before explicitly stating what this basis looks like, let's look at how our operator L behaves with respect to this manipulation of its vital space; the basis in question will fall out of this discussion in a natural way.  3.4.1  Action of I on © l (S ) 2  n  The (off-diagonal) action of L on l (V) in a particular sphere l {S ) is simply splintered up 2  2  n  into individual actions on the spheres l (S +i) and l (S -i) 2  2  n  the spheres l (S \) 2  n+  and l (S -i) 2  n  n  as given by operators going from  to Z (5 ), namely the operators gathering the parent and 2  n  children of any particular v € S or the level lowering and raising operators. Specifically for n  18  Chapter 3. The Mourre Estimate any particular v G S , let the level raising operator be denoted by n  T n - i : l (S -i) -> l (S ) 2  2  n  n  with (7r„_i0)(u)  v/2  where w G 5 _ i is the parent of v. Its adjoint is the level lowering operator n  <-i : Z (S ) -»• / ( 5 _ i ) with « _ ! 0 ) ( t ; ) = 4= 2  2  n  5 2  n  u £ S „  <£( ) u  I a„ =l u  where the it € 5„ are the children of v. So for example d a  G / (5i) and 1 {S ) 3 2  2  e  d+e  2  V2  f  a  e / (5x) 2  f +9  9 Thus for v € S  u  (L<j>)(v) = {\/2{^Q + nl)<f))(v) =  ^  0(z) which is exactly (3.10). Further-  z\a =l vz  more it is easy to verify that 7r*7r = 1. This will be a key property in showing invariance. n  Bearing the above in mind as well as the action of L on the / (5 )'s, L can be expressed in 2  n  terms of the operators {n , 7r*}, n = 0,1,2,... as follows: Let IT = ® ^ 1 ir '• l (V) —• l (V) 2  0  n  2  n  and its adjoint IT = ® £ ° < : 1 {V) -> 1 {V). Then 2  2  =0  L = v^(n + n*)  (3.11)  To see this, let v € S„, then {y/2{U + IT )<£)(«) = {y/2{ir -i + <)</>)(«) = 4>{w) + ^ n  W)  a =l uv  where w G S - i is the parent of v and the sum is taken over the u's in G SVi+i that are the n  children of v. Refer to figure 2 above. Therefore (y/2(Il + U*)<f>)(v) =  ^  W)  a s i n  z\a = l zv  Note that II maps into l (S\) © / (S ) © l (S^) © . . . , and that / (So) is contained in the kernel of n*. 2  2  2  2  2  3.4.2 Bases for l (S ), n = 0 , 1 , . . . 2  n  We are now ready to expose the bases that will do the trick in obtaining L-invariant subspaces, namely the discrete Haar bases. For the first few subspaces these are given by: 19  (3.10).  Chapter 3. The Mourre  Estimate  l (So)  / (Si)  l (S )  {1}  {^(1,1), {^(1,1),  {^(1,1,1,1), {|(1,1,1,1),  {^(1,1,1,1,1,1,1,1)  ^(1,-1)}  1(1,1,-1,-1),  ^(1,1,1,1,-1,-1,-1,-1)  75(1,-1,0,0),  1(1,1,-1,-1,0,0,0,0)  75(0,0,1,-1)}  i(0,0,0,0,1,1,-1,-1)  2  2  l (S )  2  2  2  3  (3.12)  ^(1,-1,0,0,0,0,0,0)  ^(0,0,0,0,0,0,1,-1)} where t h e digits 1,-1 are shifted b y respectively two, four a n d s i x spots, f r o m t h e fifth basis element, i n the s i x t h , seventh, a n d eighth basis elements o f l (S ).  D e n o t i n g the basis elements  2  3  of l (S )  by p  2  n  = {QnOi Qnii • • • , 0n(2 -i)}, the key t h i n g t o notice i n this choice, is that t h e n  n  first h a l f o f each of these bases c a n be expressed i n terms of the operators {7r }'s, t h a t is for n  the first few spheres: l (S ) 2  {Qoo} = {1}  0  I (Si) 2  l (S ) 2  2  {010,011}  =  W ( J ) , ^ ( l , - l ) } = {j (J,J),-^(l,-l)}  {£2o>021,022,023}  =  {*mo(l),ni{^{l,-l)),J5(h-1,0,0),  =  ^(1,1,1,1),±(1,1,-1,-1),  ?  75(0,0,1,-1)} J=(i,-i,o,0), ^(0,0,1,-1)  a n d i n general, using the II a n d II* notation f r o m the previous section we have:  n^ = 7T g nfc  n  nk  = Q( i)k a n d n+  0(n-i)fc for 0 < k < 2""  '  (3.13)  1  for 2  0 I n words, t h e effect of II a p p l i e d to Q  71-1  + 1< k < 2 - 1 n  is to move t o the corresponding basis vector 0( +i)fc i n n  nk  the next sphere, whereas the effect of II* applied t o g k is t o move to the corresponding vector n  Q(n-i)k  o  r  0 i n the previous sphere. W e w i l l use this shortly t o o b t a i n o u r desired  L-invariant  subspaces. I n general for the l (S ) we have the basis: 2  n  l (S ) 2  n  '• {0nO, 0nl,  071(2"-!), 0 n ( 2 - ! + l ) , • • • 0n(2"-l)} n  20  =  (3.14)  Chapter 3.  The Mourre  Estimate  TTn-J •  7T -j . . .7T;g(ggg), 7T _j . ..7Tg(gg ),  {•Kn-l  •••'KO(QOO),  7T _j .  ..7rj(gj(g,-i)), . . . 7T _j • • • 7 T j ( g j ( g , _ j ) ) ,  n  • • TTj ( g i i  ),  n  n  . . . ,  7T _ n  . . . ,  3  n  j (g( _ n  J)(gn-g)),  7rn-j(g(n-i)(g"- -J))) £n(2"-!)> £ n(2"- +l)' ••• > 0n(2"-2)> £n(2"-l)} J  where for any n, the last 2 0n(2»-i)  =  >  1  basis elements are given by:  n _ 1  (1,-1,0,0,0,0,0,0,..- ,0,0),  g„( n-i+i) = 2  ^(0,0,1,-i,o,o,...  2 —2 zeros  ,0,0),  2 —4 zeros  n  n  e (2»-i+2) = ^ ( 0 , 0 , 0 , 0 , 1 , - 1 , 0 , 0 , . . . , 0 , 0 ) ,  g„ 2-i+3) = ^ ( 0 , 0 , 0 , 0 , 0 , 0 , 1 , - 1 , 0 , 0 , . . .  n  (  2"-6 zeros  ,o,o),  2 -8 zeros n  g  n ( 2  n_  1 }  = ^ ( 0 , 0 , - • • , 0 , 0 , 1, "I)  2 -2 zeros n  (3.15) a n d so o n , the p a t t e r n being that of ( 1 , —1) being shifted by two spots d o w n the line for each new basis vector. I n the light of this basis, we next move o n to the L - i n v a r i a n t subspaces. N o t e that the a d d i t i o n a l basis vectors are m u t u a l l y orthogonal and orthogonal to the basis vectors o b t a i n e d f r o m the previous level by  (3.13).  3.4.3 Invariant subspaces Al*. H a v i n g the a c t i o n of L i n m i n d , as well as its expression i n terms o f the operators n = ®^_ 7T 0  a n d n* = 0 ^ L  n  n > ^ is  n O  n  o  w  possible i n a very n a t u r a l way, t o construct a n L-  invariant subspace. I n the process we w i l l see how the discrete H a a r basis is a n a t u r a l choice o f bases for the  l (S ) subspaces, n = 0 , 1 , 2 , . . . . C o n s i d e r the "constant" subspace Mo generated 2  n  b y the first basis element of each Mo  = l (S ) e n l (S ) 2  2  0  0  N o w , since 7r*7r  n  0  l {S ), n = 0 , 1 , . . . : 2  n  © 7ri7r / (5o) © . . . 2  0  =  l (S ) © 0 ~ 0  (7r • . . . • 7ri7r (/ (S ))), 2  2  = o  m  0  0  = 1, for a l l n, u p o n a p p l y i n g L — \/2(n + n*) as i n (3.11) t o Mo we r e a d i l y  o b t a i n L - i n v a r i a n c e of . M o v i a the basis element properties  (3.13), as follows:  L e t T be a n element of A^o- T h e n , T can readily be w r i t t e n out i n terms of the n a t u r a l basis f o r Mo, i.e.  i n terms of the basis {goo, gio,  Qjo, •••} = { 1> ^ ( l ) ,  f i ^ o U ) , • ••>  TTJ ... 7ro(l),... } as, T = ao • 1 © a i • 7ro(l)ffi . . . where a j € C so that T is represented i n this  21  Chapter 3. The Mourre Estimate  basis by (ctQ, ct\, a ,0:3,...). We next apply L = \/2(n + II*) to T, and since 7r*7r = 2  n  1  for all  n obtain: LT = v^(a 7ro(l)eai7ri7ro(l)©...) + v2(ai7r*7r(l) © a7r*7ri7r(l) © ...) /  0  0  2  0  = \/2(ai(l)ffi(o-o + a )7r (l)ffi(ai + a )7ri7r (l)ffi...) 2  0  3  0  which, as can be seen, is an element of Mo and is represented by y/2(ai, (ao+a^), i i + s), • • •) a  a  in the above basis of Mo- Thus the subspace Mo is invariant under L and when restricting L to this subspace, L is unitarily equivalent to a discrete Laplacian acting on a half-line, namely: (LT)(n) = v^(T(n + l) + T(n-l))  ={  y/2{a i + a -i), n+  for n > 1  n  %/2ai,  for n — 0  where T(n) is the value of T with respect to the n basis element of Mo (hence it's n th  th  coefficient). The other invariant subspaces Mk are obtained in a similar fashion, namely by picking all the second basis elements of each l (S ) to form Mi, then all the third basis elements to 2  n  form M2 and so on. The key difference being the starting point, or starting space of each of the subspaces. In particular, the starting space for the next one in line, Mi, is generated by the second basis elements of l {Si), since the only basis element of 1 (SQ), namely £00 was used 2  for Mo- Let Q  ntk  2  =span(e„ ), e.g. Q ,o =span(£ o) and Qi,i =span(e ) = span(^=(l,-1)). fc  0  0  u  Then up to date we have: l (Sp)  l (S!)  l (S )  Q ,o  ffi 7T(Qo,o) ffi  7ri7ro(Qo,o)  ffi  7Tl(Ql,l)  ©  2  M: 0  Mi:  0  2  2  2  0  Q  I (S )  2  ©  U  3  WiMQofi)  © •••  (- ) 3  16  7T27Tl{Q l) ffi . . . h  Continuing in this line, we see that since the two vectors g(l, 1,1,1) and -(1,1, —1, —1) have already been used up in defining Afo and Mi- each of the remaining basis vectors of / (5 ), 2  2  7|(1) -1,0,0)) and ^=(0,0,1, -1)), will generate the starting spaces of M2 and M3, namely Q.2,2 =span(^=(l,—1,0,0)) and Q 3 =span(^=(0,0,1, —1)), such that(3.16) now reads: 2)  22  Chapter 3.  The Mourre  Mo-  Estimate  Qo,o  ©  TTO(QO,O)  Mi:  Qi,i  7ri7ro(Qo,o)  ©  ©  © 7r27i"i7ro(Qo,o)  7ri(Qi,i)  ir iri(Qi,i)  ©  2  © •••  ©  ... (3.17)  M  ••  2  Q,  ©  7T2(Q2,2)  ©  . . .  Q ,3  ©  7T2(Q2,3)  ©  • • •  2 2  M: 3  2  Looking back on the basis elements in (3.12-15), notice at this point that the starting space of each of M\, M and M3 is generated by one of the basis elements in the last half of the bases 2  of 1 {S ) and ^(S^) not previously used in the definitions of Mo and M\. 2  2  This is generalized in constructing the subspaces Mk- Namely the procedure involved in their construction simply consists in picking one by one, each of the last 2 Qnk £ l (S )i  2  2  n  n_1  basis elements,  < A; < 2" — 1, not used in defining a previous subspace, as generating  n_1  elements of the starting spaces Q ,k of Mk and this for each n. Hence in general, employing n  the 7r notation again, we have that n  00  Mk - Qnk © Kn(Qnk) © ^n+l^n{Qnk) © • • • = Qnk © @ l—n  • • • ^n+\^n{Qnk),  where [log(A; + 1)] = n and Q k = span(g„k) as starting space. 2  n  As for L-invariance of Mfc's, this follows in the same manner as before; let {g k, Q(n+i)ki •••} n  be an orthonormal basis for Mk and let T G Mk- Then T can be written in terms of this basis, and is represented say by (a fc, a( +i)fc> (n+2)fc> • • •) a  n  n  m  t  m  s  basis. Thus L T after calculation  is represented in the above basis by v^Q^n+i)*;, nk + a(n+2)fc> &(n+i)k + (n+3)k, •••)• So we a  a  have shown that under this decomposition of 1 {V): 2  Proposition 3.1 The discrete Laplacian L when restricted to the invariant subspaces Mk of l (V) 2  is unitarily equivalent to a one dimensional discrete Laplacian acting on a half-line, that  is to C : 1 {Z ) -»• 1 {Z ) as given in (3.6). 2  +  2  +  Corollary 3.1 L can be written as a direct sum of its restrictions to the invariant subspaces Mk, that is , L- 0 ^ L \ o  M k  - 0°^ C o  k  where C = L\ k  23  Mk  Chapter 3. The Mourre Estimate Proof  If we line up the the corresponding values of L\M on typical elements K  GM  for  K  k = 0 , 1 , 2 , 3 , . . . as follows: •^IA^O^O =  y/2(o!io, o o + o o, «io + 0  2  030,  •••)  L\M 1^1=  y/2(Q,  o ,  011+031,  •••)  LU* T =  v^(0,  0,  a ,  ...)  a  2  21  32  then we see that by taking a direct sum of the values of L on each subspace we get back the original value of L in terms of the values <f>(v), v G V, namely \/2oio(l) =  4>{ \)+4>{ 2)-, v  V  where v\  and V2 are the children of the initial node VQ, \/2((ooo+o o)0io+O2i0ii) = <f>{v3)+4>(v4)+(f)(vo), 2  where w and V4 are the children of v\ and vo its parent, and so on. o 3  We now see that working on the half-line is a basic step, hence a building block, for working on a binary tree. With this decomposition at hand, we are now ready to consider the Mourre estimate for L acting on 1 {V) where V is the set of vertices of a binary tree. This is 2  the first step to showing that a Mourre estimate holds for our Schrodinger operator H — L + Q acting on l (V). 2  Recall our notation from previous sections: C : /  2  ( Z ) —> Z (Z+) +  2  is the half-  line Laplacian, and L : l (V) —> l (V)) is the binary tree Laplacian. By Corollary 3.1 we have 2  2  that L above is unitarily equivalent to a direct sum of half-line Laplacian operators C, namely = © A and by (3.9) we have that the half-line Laplacian  L = ®L\M  K  C  satisfies a Mourre  estimate. Hence, we will want to use this fact to obtain the Mourre estimate for L acting on 1 {V). We start by discussing the form of the appropriate conjugate operator and then proceed 2  to use it to show that a Mourre estimate holds.  3.4.4 Definition of the conjugate operator A The first definition of the conjugate operator associated with L acting on 1 {V) we can use is 2  A=0  f c  Ak, where Ak is the conjugate operator associated with the half-line Laplacian C , as k  defined in (3.8). This, we use in the next section, to show that the Mourre estimate holds for L acting on 1 {V). But we would like to find a more tangible definition of the conjugate operator 2  24  Chapter 3. The Mourre Estimate A, such that A\M = Ak- Is this possible ? The answer is yes; let k  A =-(U*(2(X- N) + 1) - (2(X -N) + 1)11) = I ( A * - A),  (3.18)  l  where X<t>(v)  = |t#(u)  and where Ng  nk  = X g k = [log (^ + l)l£nfc for g k the k  n  2  basis  n  vector of the space l (S ) and \x] the smallest integer > x. Let g fe G A ^ , then Xk is 2  n  n  a number representing the starting position from the origin of the space Mk, and A\M  1S k  unitarily equivalent to A = Akforany A; > 0. For, let Tfc be represented in MkS basis by Kfe,  "(n+i)*, Oi{n+2)k,  a  ,...)  [n+z)k  then AT = | ( n * ( 2 ( X k  = 5((3a( +i)fc,5a( 2)A; - 3a ,7o;( 3) - 5 a n  n  n+  n+  fc  (n+1)fc  - iV)  + 1) - (2(X  — AT)  + l)II)T  fc  , . . . ) ) which is in Mk and is exactly (3.8)  that was derived in section 3.3.3. for the action of A. Hence, we also have that  Proposition 3.2 Let A = %{IT(2(X - N) + 1) - {2{X - N) + 1)11) : 1 {V) ->• Z (V). 2  A|^  f c  2  T/ierc  is unitarily equivalent to the half-line conjugate operator A and A = ® Ak — ®/; A\M fc  k  Further it is also possible to find an expression of (A<f>)(v) in terms of the values of the function 4> at each and all of the sites in the next and previous spheres. We state the expression here, and refer the reader to appendix A for the derivation. Specifically, for v G S , we have : n  MM  =^ X > M 1=0  where k — . ^ ., , m 2^ )' n  n  =  f  E  +  \ »€S„  , \wo> 2^ 'I  n  1=0  /  +1  =  E  * M )  \ «ues„_i  (- ) 3  19  /  M * the level at which u is located, ~f(v,w) s  represents the youngest common ancestor of v and w (possibly equal to v or w). The inner sums in (3.19) just sum up the contributions of all of the vertices in the successive and preceding levels; in the first inner sum, for / = 0,... , n — 1 there are 2 ~ terms and for / = n there are n  l  2 terms, while in the second inner sum, for Z = 0,... , n — 2 there are 2 ™ ' -  -2  terms and for  I — n — 1 there is one term. The coefficients a i > 0 as well as b j > 0 depend on n and I and n>  25  n  Chapter 3. The Mourre Estimate are given by 1  for / = 0  2n + 2 + E*=i(2n - (2fc - \))2 ~ - (2n - (21 - 1))2'~ k  x  1  for f = 1,... n - 1  n—1  o  n + - + VJ(2n-(2fc-l))2 k=o fe  for / = n  1  (3.20) and for I = 0 2  b ,i = { n  " + El=\(2n - (2fc + l))2  fc  - - (2n - (2/ x  + l))^"  1  for / = 1,... n - 2  n-2  + - + ^ ( 2 n - ( 2 A + l))2 fc  (3.21)  for I = n — 1  1  ;  fe=0  For our purposes though it will be sufficient to notice that  2  (]T v • 2""'  + 2o ) BlB  =l  2(n+1) (  (n+l)+2"  }  _l_  =  (2n+ 3 ) = c i ( n ) = 0 { n )  { 32 2 )  V2'  and n-2  (J2( n,l^ - - ) b  n  l  + bn,n-l) = k  2  r  L  J  - ) - ^ + ^ c  2  ( n )  = 0(n)  (3.23)  where O(n) denotes a quantity which satisfies |0(n)| < Cn for C a constant. In the next section we show that the Mourre estimate holds for the binary tree Laplacian. Since each A is selfadjoint, A is as well, hence conditions 1 and 2 of the Mourre estimate follow k  as usual. It remains to verify the third condition.  3.4.5  The Mourre estimate for L acting on l (V) 2  Having defined our conjugate operator, the key elements in showing that the Mourre estimate, specifically the third condition, holds for L acting on l (V), are: 2  • @ E (C ) k  I  k  = E (@ C ) I  k  k  =  E (L), I  • ©fc[£fci-4fc] = [0^ Ch, © f c A ] = [L,iA] by invariance of the subspaces M  k  and A , i.e. Propositions 3.1 and 3.2. k  26  under C  k  Chapter 3. The Mourre  Estimate  Then starting with the Mourre estimate for each of the A's and using the above obtain: M/k]S/(/: )) >  ® 9E](£ )  fc  k  k  where 0 = min{4sin(i)}, that is, 2  tei  0 * E (C ) ®[C ,iAk]® Ei(Ck) r  k  k  >  k  0® Er(C ) k  0E](L)  £/(L)[L,L4]£/(L) >  k  (3.24)  The second step then consists in showing that the Mourre estimate for H = L + Q holds with certain conditions on Q. This is done in the next section.  3.5  Mourre Estimate for H acting on l (V) 2  So, L acting on 1 {V) satisfies the Mourre estimate. What about the Schrodinger operator 2  H = L + Q? If Q is such that • q(v) —> 0 as \v\ —> oo; all we ask here is that the potential decrease at infinity in any case, and either • q(v) decays faster than -p-- and/or • the differences \q{v) — q(w)\ decay faster than  M  successive sphere, that is in more general terms, if Q — Qi +  for w any vertex in the preceding or  where Q\ is such that Hgi(v) —> 0 as \v\ —»• oo  and Q2 is such that |vj|g2(^) — Q2(w)| —• 0 as \v\ —> 00 for w any vertex in either the preceding or successive spheres and qi(v) —> 0 as |v| — > 00, then we will see that H satisfies the Mourre estimate with the conjugate operator A given in (3.19). How so? Since we were able to nicely write L as a direct sum of certain operators satisfying the Mourre estimate, it is a natural question to ask whether the same thing can be done for H, specifically for Q? The answer is no. When we introduce a potential Q and consider H instead of L things are different, in that the subspaces M are not invariant under the potential. Hence, k  27  Chapter 3.  The Mourre  Estimate  Q can not be written as a direct sum of "component" operators. Remark: It is interesting though, to note that we can nicely express the block-diagonal terms as some type of convolution operator, when projecting Image(Q) back onto the M^s.  We will have to bear the rebellious  behavior of Q in mind, in the following lines. Let's show that H satisfies the three conditions of the Mourre estimate, one by one : Condition 1: D{A) D D{H) is dense in D(H) in the norm \\(H + i) • || Since q(v) -> 0 as \v\ —>• oo implies that H — L + Q is bounded and since A is selfadjoint condition 1 follows in the same manner as before. To show that the following two conditions hold, we will use the important fact that [Q, iA] is compact, which we will show in the next section (Lemma 3.1 and/or Lemma 3.2), to not interrupt the chain of thought. Condition 2: The form RH(i)[H,iA]R (i)  is bounded, where il#(i) = (H — z )  H  - 1  is the  resolvent corresponding to H. If [Q, iA] is compact, it is in particular bounded and since [H, iA] = [L,iA] + [Q,iA], condition 2 follows from condition 2 for L and boundness of H. Condition 3: If I is a sufficiently small interval containing A, then Ej(H)[H,iA]Ej(H) 6Ei(H)  2  >  + K where 6 > 0 and K is a compact operator. Definitions of Ei() and Ej(H) as  usual. Note that it is sufficient to consider smoothed "cutoff" spectral projections we prove that  (H)[H,iA]E^(H)  > 6EJ \H) S  2  EJ*\H).  For if  + K then conjugation by the sharp spectral  projection Ej(H) on I C J, will yield the appropriate estimate given in condition 3. The idea is to use the equality [H, iA] — [L, iA] + [Q, iA] as well as the Mourre estimate for L . We have: Ef{H)[HM]E^{H)  = Ef{H)[LM]Ef(H)  28  +  Ef{H)[Q,iA]E j\H) {  Chapter 3. The Mourre  Estimate  Then the second term on the right-hand side is compact (call it K~o) for [Q,iA] compact. It remains to convert the first term into 6EJ \H) . S  Notice the spectral functions E^(H)  2  are in  terms of H and not L. So we can't pull out 0 directly as was done in the free case. This is where we use the Mourre estimate for L, i.e. E j (L)[L,iA]E j { S)  { S)  (L) > 6EJ \L) . S  A couple of  2  intermediate manipulations are in order, though: • rewrite E j {H) as (E J \H) { S)  (  S  - E J \L)) (  S  + E j (L) on both right-hand and left-hand sides { S)  of the first term above. • In order to obtain compact terms, when reducing E*J\H)[L,iA]E^(H) Ej \L)[L,iA]Ej \L), S  to  use the abstract Stone-Weierstrass theorem :  S  Let Coo(]R) = {/ : lim f(x) = 0, / continuous} with sup norm. Suppose S C Coo(M) |x|—)-00  satisfies: (i)  S is norm closed  (ii)  S is a vector space and an involutive algebra, i.e. f,g G S => af + PG G S where a, (3 G C and f G 5, f g G S  (iii)  (x — z)  G S for some z G C, Rez ^ 0  -1  Then S = Coo(R)o. So, let S = '{/ G Coo(E) : f(L) - f(H) is compact}. Then we Claim S is sup-norm closed and an involutive algebra; Proof Let f  n  G S converge to g, then ||(/„(!) - f (H)) N  g(L)\\ + \\g(H)-f (H)\\ N  f*(L)  => (g(L)-g(H))  - (g(L) - g(H))\\ < \\f (L) n  compact, J(H) = f*(H) hence J{L) -J(H)  =  — f*(H) = (f(L) — f(H))* is compact, as the adjoint of a compact operator is  compact and fgES  i.e. f(H)g(H)  f(H)(g(H)—g(L))+g(L)(f(H)-f(L))  - f(L)g(L) is compact as f(H)g(H)  — f(L)g(L) =  which is compact since both / and g are elements  of S. Finally as will be shown in a lemma in the next section, Q is compact, hence RH( ) Z  —  RL(Z)  = RH(Z)QRL(Z)  is compact and we are done.  So S = Coo(R) and  (E j {H) - E j {L)) is compact o ( S)  { S)  • Use the Mourre estimate (3.24) obtained for L in the previous section. Specifically we have, 29  Chapter 3. The Mourre Estimate B y the first two steps : Ej (H)[H,iA)E J {H) )  i  ( S}  Q  X  { S]  2  and K = (E^(H)  E j\H)[L,iA](Ef{H)-E j\L)) {  where K is given above, K =  = K + K + K + E j {L)[L,iA]E j (L)  l)  {  0  -Ej^(L))[L,iA]E^(L)  2  x  are compact by  the abstract Stone-Weierstrass argument. B y the two last steps : We repeat here for completeness parts of the argument used for L in (3.24);  E j (L)[L,iA]E j (  S)  { S)  (L)  Ki — # ( ( £ ^ ) ( . L ) — (Ej ^) (H)) 2  S  2  =  e ^ ; ' ^ ) ^ ^ ] ^ ^ * )  >  0 . 0(E j ) (£ )  >  K + eE j\H)  { S)  = e{Eff{L)  2  k  for 9 = min{4sin(*) } 2  where  {  A  is compact by the abstract Stone-Weierstrass theorem stated  above. 3.5.1  R a t e of decrease of the "differences" of Q on l (V) 2  In this section, we use the discrete tangible definition of A as was stated in (3.19) to show that, under certain conditions on Q, [Q,iA] is compact. What we obtain is an expression for ([Q, iA]4>)(v) = (B<f>)(v) in terms of the values of the functions <> / and q evaluated at each and all of the sites in the previous and next sphere. Specifically for v € S , breaking up (3.19) a bit n  differently, and letting £(v,w) — (q{w) — q(v)), we have:  {[QMWiv)  =  y (  - ~Y f  E  C K ^ H + XX*  C(v,w)(j)(w) + Y^ "l [  E  b  V u>€S_i n  f(v,w)eSo  1=1  (  E  E  V ™€S _! n  -y{v,w)eSi  C ( ^ ¥ H j )  C(v,«>)0M ) )  (  3  25  )  / /  Further this expression can be recognized as an expansion of (B4>)(v) in terms of the {5 } = VW  {dij} basis. That is, if B is thought of as an infinite matrix then its matrix elements in this basis are given by the corresponding coefficients in (3.25). Now, we want compactness of B — [Q, iA].  Lemma 3.1 Suppose \q(v) — q(w)\ is of order o(\v\~ ) for allv andw such that | \v\ — \w\ | = 1, l  then i) Q is compact, ii) B — \Q,iA\  is compact. 30  Chapter 3. The Mourre Estimate Proof  W e proceed to prove the harder of the two ii), the former i) is proved i n a similar  fashion.  P  Let \\B\\  n  (B" l (Sj). 2  be the projection onto  \\B - P B\\ =  We wish to show that  =0  n  —>• 0 as n —>• oo, for then B is a limit o f finite rank operators, hence compact. L e t _  I  Bv,w if  \v\ > n  ./ ^ ^2 \Bi,^\ < M\(n)  . T h e n , by Schur's lemma, i f sup„  if  0 I^Sl  (3.22)  and  <  M  (3.23)  W  and  \v\ < n  2 ( n )forn = \v\ then  < (Mi(n)M (n))2.  B u t b y using formulas  2  o f the previous section a n d by assuming that, for any  \q(v) — q(w)\  preceding or successive spheres, the differences following: F o r convenience let \q(v) — q{w)\ =  ^  are of order  w  i n either o f the  o(n ) _1  we o b t a i n the  i n the next calculation, t h o u g h keeping i n  m i n d that other forms of \q(v) — q(w)\ of order o(|u| ) are also possible. _1  supJTlsftJl <  £  Y  f v  w:\w\>n  w  ( XX>< ( E M \ 1=0 \ w£S i j(v,w)eSi  V  '  n+  + y ( D v (  E  !  W  )\  J + ^,n E K(V,W)\ / w£S J i(v,w)es a  n+l  n  IC(«,w)l ) + Kn-i  y{v,w)&Si  E KMl)  7(v,a))eS _i n  w:\w\>n \/2 <  -£-(6n  C + 7)^7  = M i ( n ) by  In a similar fashion we o b t a i n that  M (n) = 2  0{n- ) £  and  (3.22)  and  V^-B^l w —'  sup  < (Mi(n)M (n))a = 2  (3.23)  of the previous section.  < -^(6n + 7)-^— = M (n). HenceMi(n) 4  0(n~ ) e  n"" 1  1  -» 0 as n ->  6  00.  2  T h u s 5 = [Q,»A] is  the limit o f finite rank operators, hence compact o  A l t e r n a t i v e l y this c a n be shown b y using the direct s u m formulation of A. T h i s is done i n the next subsection.  31  Chapter 3. The Mourre Estimate  Rate of decrease of Q itself  3.5.2  Recall that in analogy to L being a direct sum of operators L\M , where the A-ffc's are invariant K  under L, we had established that A = ®kAk, where Ak is the conjugate operator acting on the subspace Mk and is unitarily equivalent to the half-line conjugate operator A. We wish to use in some way the Mk subspace decomposition of l (V). We have that the Mk's are invariant 2  under Ak but not under Q. This is troublesome. So introduce a potential Q, for which they are. In particular, for v G S let <Q$(v) = n  since q(v) -> 0 as \v\ —> oo and Q =  max {v\v e  • (j>(v) so Qc/>(v) -» 0 as \v\ —> oo S} n  = ®jQj- This potential enables us to use the  ®JQ\MJ  Mk subspace structure by considering QQ~ QA instead of QA. Hence compactness of QA is 1  reduced to boundness of QQr and compactness of QA. 1  By definition of Q, Q Q  _ 1  is bounded. Also since Mk is invariant under Q,  QA = (®jQj)(® Ak) k  = ®S QjA jk  =  k  ® QkA k  k  Firstly we wish to show that QkAk is compact. Recall that Ak — |(n*(2(A" -N) + l) — (2(X N) + l)n). Let Bk = QkAk- We omit the | factor in the next calculation, for convenience. But B  k  =  Q Il*{2{X-N) + l)-Q {2{X-N)  =  2(Q (X - N)U* + Q [U*,X] + Q [N,U*]) + Q U* - 2Q (X - N)U - Q U  k  + 1)U  k  k  k  k  k  k  = 2(Q (X - N)(U* - n) + Q [U*,X]) + Q (n* -U)asN k  k  fc  k  commutes with n*  Since the multiplication operator X — N satisfies (X — N)p k = (n — \k)Qnk < Qnk = n  n  Xp k, recall that n  was given by flog (A; + 1)], and since the operators ((n* — Il)(f))(v) = 2  -^=( y~] <j>(u) — 4>{w)) where w e 5 _ i and ([U*,X]4>)(v) = n  Q'tiV — 1  seen bounded, if  Qiity  <t>{u), are as can  X  —> 0 as \w\ —• oo, all terms are compact and hence so is Bk by a finite  rank argument as in previous section, i.e. Secondly we wish to show that  ||-BJJ. ^|| n  —>• 0  as n —>• oo.  0 as k ->• oo. But, ||5A;|| < 2 C\ sup(|u;||g(u;)|) + (2 C2 +  C\)sup(|q(io)|), where the sup takes place over w €  supp g k and C\ = nt  — H\\ and  C2 = ||[n*,X]||. Thus recalling that as k -» 00 the starting point of the subspace Mk moves 32  Chapter 3. The Mourre Estimate further and further away from the origin, we have that as k ->• oo \w\ -> oo, thus by condition on Qk as \w\ ->• oo, we get that Ve, 3 M such that ||Bfc|| < e for \w\ > M, for all k. Hence \\Bk\\ ->• 0 as k -> oo. Having established that the JS/t's are compact and that  —>• 0 as A; -» oo, we have that  ®kQkAk is compact, hence QA is compact. Since QQ~ is bounded and QA is compact, we X  have that QA — QQ~ <QA is compact 1  [Q,iA] is compact. We have shown that:  L e m m a 3.2 Suppose that max {|<?(v)|} = o(n~ ). Then [Q,iA] is compact. v es 1  n  We proceed in the next chapter, in using the fact that the Mourre estimate holds for our Schrodinger operator H = L + Q with conditions of Q as stated in either of the above sections, to obtain asymptotic completeness. In the process we will also encounter the limiting absorption principle.  33  Chapter 4 Asymptotic Completeness  This chapter concerns a couple of applications springing from the Mourre estimate. The goal of this chapter is not to go in the details of the proofs but to give the reader a brief overview of how asymptotic completeness can be achieved via Mourre theory. Specifically we will deal with the limiting absorption principle (LAP) as a means to prove asymptotic completeness. Let's start by discussing asymptotic completeness a bit more, then see how the L A P can be used.  4.1  Asymptotic Completeness Je vous demande: What is captivating when dealing with many-body problems ? It is  exactly this many-body aspect. Is it possible to grasp a mathematically precise picture of an arbitrary number of bodies interacting and scattering via short range or long range (potential) interactions for example? In scattering theory, this is where the notion of asymptotic completeness comes in. Recall from the introduction that intuitively a physical system composed of an arbitrary number of particles is said to be asymptotically complete if at large negative and positive times, the system is decomposed into bound clusters evolving freely. A cluster may be composed of only one particle. In the two-body problem in non-relativistic quantum mechanics, where the underlying Hilbert space for either of the particles is L ( R ) , one can 2  3  remove the center of mass of the system and reduce the problem to the one-body case. Thus the one-body problem in an external field though less fantastic remains a first step. In the next lines we scrutinize asymptotic completeness under a mathematical magnifying glass. What we need to start off are the notions of scattering states and bound states. Intuitively 34  Chapter 4. Asymptotic Completeness a bound state is one such that all subsystems stay confined to a bounded region for all time and a scattering state is one such that the subsystems leaves every finite bounded region, say a finite ball of radius r as t —>• oo and as t ->• -oo. What naturally interest us, are the scattering states, not the static bound states! Seems simple enough. These states naturally live in a Hilbert space, in our case in 1 {V). But as we saw in the previous chapter 1 {V) can be decomposed into 2  2  Tipp®T-L . The bound states naturally live in Ti . Let S C Ti be the set of scattering states. A c  VP  c  priori we do not know whether S = Ti . This depends on the mathematically rigorous definition i r of a scattering state; if we let S = {/ € H \ lim — / \\XrC~ f\\ dt = 0 Vr < oo}, where c  T  lHt  {  1  if \x\ < r  0  if \x\ > r  2  , then Ruelle's theorem tells  us that if / 6 Ti , then / € 5, so that indeed S — Ti . In words Ruelle's theorem states that c  c  in the time average, the states in Ti leave each bounded region, hence are scattering states in c  our previous sense. We note here that there are only scattering states in the free case, i.e. all elements of the Hilbert space Ti. Now how does asymptotic completeness fit into all of this? To answer this we need to introduce, among other thing, the notion of wave operators, fl±, which "keep track" of these scattering states. In the case of a Schrodinger operator H = —L + Q this involves the interplay between the free and perturbed dynamics of the system. We will see that if these wave operators exist and their range is equal to the space of scattering states, S, then asymptotic completeness holds. So asymptotic completeness reduces to existence of operators and showing that their range is in fact Ti . c  4.1.1 Construction of wave operators Let e~  lLt  and e~  be the unitary one-parameter groups associated to the free and  lHt  perturbed systems respectively. Under the notation above, S is a strongly closed linear set that is invariant under e~  lHt  and the projection P (H) can be associated to it. To translate C  asymptotic completeness in mathematical terms what we want is an approximation of the total evolution of the system by a free evolution (and vice-versa) as t —>• ±oo. There are two parts to  35  Chapter 4. Asymptotic Completeness the definition, i n v o l v i n g a n a p p r o x i m a t i o n i n the distant past of a free state m o v i n g under the free d y n a m i c s (so a prepared state), by a scattered state m o v i n g under the p e r t u r b e d d y n a m i c s a n d a n a p p r o x i m a t i o n i n the distant future of this scattered state m o v i n g under the p e r t u r b e d d y n a m i c s b y a free state m o v i n g under the free d y n a m i c s .  M o r e precisely, i f we are given any freely evolving state element e~ 4>- where 0_ € So lLt  represents the l a b prepared state at t — 0 (it doesn't make sense at least to this d a y t o start w i t h <p ) we d e m a n d that their exists a state element 4> E S m o v i n g under the p e r t u r b e d d y n a m i c s +  such that  e~ 4> — e~ 4>- A 0 as t -> —oo. Further, we also d e m a n d that s t a r t i n g f r o m this lHt  tLt  state element 0 E S m o v i n g under the p e r t u r b e d d y n a m i c s , there exists a freely evolving state  4>+ E% such that e~ (f> — e~ 4) xHt  lLt  +  s-  -4 0 as t ->• +oo. I n other terms:  l i m \\e- </>-e- <t>-\\ = 0 a n d siHt  l i m \\e' 4> - e~ (j) \\ = 0  iLt  iHt  iLt  +  F r o m these statements we can extract the wave operators we are after. W e c a n define these, u s i n g u n i t a r i t y of  e~ a n d e~ , by: lLt  Q-(H,L)=s-  lHt  l i m e e~ iHt  iLt  a n d Q (L,H) +  t—>—oo  = s-  lim  e e~ V {H) iLt  iHt  c  t—>+oo  G e n e r a l i z i n g this t o b o t h directions i n time for the sake o f m a t h e m a t i c a l s y m m e t r y gives i n fact four wave operators, £l±(H,L)  a n d Q±(L,H).  A s y m p t o t i c completeness holds i f these  four wave operators exist. E q u i v a l e n t l y , a n d of more p r a c t i c a l use a s y m p t o t i c completeness is equivalent t o existence of  £l±(H, L) = Q± a n d equalities Ran(f2_) = Ran(fJ+) = S = ri , the c  last equality following as noted above.  M a n y approaches t o proving asymptotic completeness for m a n y b o d y problems i n t h e continuous case w i t h different conditions o n the potential exist i n the literature t o d a y ; see for example [Gr], [SSo], [En] a n d m a n y more. O n e line of thought among others to prove a s y m p t o t i c completeness, is t o start by showing that the M o u r r e estimate w i t h a n added f o u r t h c o n d i t i o n holds for the operator considered. T h i s then implies what is k n o w n i n the literature as t h e l i m i t i n g a b s o r p t i o n principle ( L A P ) , w h i c h we w i l l discuss briefly i n the next section. I n t u r n the L A P not o n l y implies absence of singular continuous s p e c t r u m (so 5 =  36  T-Lac) b u t also t h a t  Chapter 4. Asymptotic Completeness Q / 1  2  is "smooth" with respect to H in some sense which will also be briefly discussed; this  implies existence of the four wave operators hence asymptotic completeness.  4.2 The limiting absorption principle We start by putting forth this fourth condition to the Mourre estimate, proceed to show that our Schrodinger operator H = —L+Q satisfies it, then discuss the limiting absorption principle.  4.2.1  Added fourth condition  Thus far, we know that for our Schrodinger operator H = —L + Q, the Mourre estimate holds. The fourth condition needed to get the L A P is as follows: 4) The form [[#,zA],iA] is bounded.  C l a i m The Schrodinger operator H = L + Q with the associated conjugate operator considere in chapter 3 satisfy this fourth condition. Proof Since by condition 2) for L and A acting in l (V), we have that  A] is bounded as  2  [£, iA] = 4sin (i) for C and A acting on the half-line and since [[£, iA], iA] = —8sin (r.) cos(i), 2  2  we have that [[L,L4.],iA] is bounded. It remains to show [[Q,iA],L4] is bounded. There are several approaches to this: i) Suppose Q is decaying faster than —77. Since [[Q,iA],iA] = \W —QAA + 2AQA — AAQ by a similar argument as in Lemma 3.2, this will assure compactness hence boundness of each term, ii) Using the definition of A in (3.19), and letting £(u, w, z) = 2q(w) — q(v) — q(z), p = n + l and r = n — 1, we obtain for v G S , that n  [[Q,iA]M]<i>(v) =i  yJEv E \ 1=0  P~  l  {^rE P>*( A  ™es  ^  fc=0  i(v,w)eSi n+1  v  ^ :es  n  n + 2  J 2  -y(w,z)eS £  C(«, « ; , * ) * ( * ) )  ^ *£S -y(w,z)eS  -  '  n+2  2  /n-1 V/=0  )  J  «es _! n  ^ f ^ b J  k=Q  k  1  y(v,w)€Si  k  (  -  '  ~f(w,z)es  \ ^\ *es _  C{v,w,z)(j)(z)\ k  /  k=0  E  £  fc=o  C(v,w,zMz))\  ^ z€S _ i(w,z)eS n  2  ' >/  k  Let D = P [[Q,iA],iA]. By assuming that \((v,w,z)\, i.e. the difference \2q(w) - q(v) - q(z)\ n  37  Chapter 4.  Asymptotic  Completeness  decays faster than j — ^ and by using a similar argument as in Lemma 3.1, we obtain that \\D\\< (A n  2  x  + A n + A ) ^ — = 0(n~ )  which -> 0 as n -)• oo o  e  2  3  (4.1)  We refer the reader to the second appendix for further details. So we see that [[Q, iA], iA] is compact hence bounded. It will be sufficient for our purpose to know that if this condition holds then the limiting absorption principle for our Schrodinger operator can be derived. See [CKFS] for further details. Nevertheless let's discuss the L A P a tiny bit.  4.2.2  Classical and less classical approaches to the LAP In its abstract and most general terms the L A P deals with limits of the resolvent RH(Z) for  a self-adjoint operator H, as z approaches the real axis. The question it answers is : does RH{Z) remain bounded (i.e. assume "boundary values") on the positive real axis for Im(z) —>• Ol ?The L A P expounds the optimal topology in which the above limits exist. Since the real positive axis contains o~(H) such limits do not exist in terms of operator norm convergence in L ( R ) . One 2  n  approach is to consider RH(Z) as an operator valued mapping from a suitable Banach space with values in its dual. In particular in the continuous case, these suitable Banach spaces turn out to be given by L < (R ) = {/(x) | (1 + |rc| ) / /(x) € L (R } and the mapping an element 2  s  of e.g. B(L ' ,L ~ ), 2 S  to L ' (W) 2 S  2  n  2  s  2  2  n  or even an element of B(L ' , 7l ~ ) where L ' ( R ) is the dual space  S  2 S  and U ~ = {f(x)\D f 2  s  a  € L ~ }, 2  s  2  s  2  _ s  n  0 < \a\ < 2 for D taken in the distributional a  sense. This is the classical approach to the LAP. See [Ag] or [BD] for more details. The L A P gives us under suitable conditions on the operator: lim  sup  \\R(z)\\ ( 2,. 2,-. <C B  L  tH  )  We refer the reader to [BD] for an in depth discussion of the LAP. Another approach relies on considering Hilbert space limits of BRH(Z)B  for an appro-  priate operator B. The use of the operator B "replaces" in some sense the notion of weighted space. As to be expected the assumptions used in this version of the L A P will differ from the classical assumptions. They amount to the Mourre estimate with the added fourth condition! We refer the reader to [PSS] for further details. This version of the L A P reads as follows: 38  Chapter 4. Asymptotic Completeness  Theorem Suppose H and A satisfy the Mourre estimate on some interval I with the added fourth condition. Then the set of eigenvalues of H is discrete in I and sup  ||(|A| + l ) - ( i f - A - i ) - ( | A | + l ) - | | < o o a  1  (4.2)  Q  M  0<H<\  for any fixed a > 5. Equation ^.2 holds uniformly as A runs through compacts of I\D where D is the set of eigenvalues. Since the hypothesis holds for our Schrodinger operator H = —L + Q, we have this version of the L A P at our disposal.  T h i s result w i l l be used i n the next section to o b t a i n  l o c a l i J - s m o o t h n e s s of the operator Q% w h i c h implies existence of the wave operators, hence a s y m p t o t i c completeness, since we already have absence of singular continuous s p e c t r u m , by the M o u r r e estimate. L e t ' s start though by this notion of .ff-smoothness.  4.3  if-smoothness  Definition 4.1 Let H be a self-adjoint operator with resolvent RH{Z) = {H — z)~ . l  Let B  be a closed operator. B is called H-smooth if and only if for each 4> E H and each e ^ 0, RH(^ + it)4> S D(B) for almost all A € M and moreover 1 r °° —j / (\\BR (X + ie)<j>\\ + \\BR (X - ie)<j>\\ )d\ < 00 +  \\B\\ = 2  H  sup IMI = l  2  2  H  4  7  r  J  H  ~°°  e>0  That  =  sgn(Q)|<32| i f f - s m o o t h follows from Q^RQ* b e i n g b o u n d e d by T h e o r e m s  X I I I . 3 0 i n [RS3]. T h u s by T h e o r e m X I I I . 2 4 of [RS3] this implies existence of the wave-operators, hence a s y m p t o t i c completeness. L e t ' s see how the L A P is used to show t h a t QzRQ^ is b o u n d e d : S t a r t b y r e w r i t i n g Q^RQ^  as :  QiRQk = Q5(l + \A\) \l + \A\)- R(l + \A\)- \l + \M) Q^ * v ' * * ' +Q  B y the L A P , (1 + over Im(z)  a  a  +a  \A\)~ R(z)(l + \A\)- is b o u n d e d for 0 < Re(*) < 1, a > \ a n d u n i f o r m l y a  a  r u n n i n g t h r o u g h compact sets i n the complex plane, take away the eigenvalues a n d  39  Chapter 4. Asymptotic Completeness the thresholds. Hence we need to show that Q^(l + \A\)  and (1 + |A|) <2^ are bounded.  +a  +Q  Let's show that the latter is bounded: Rewrite it as (1 + | A | ) Q 5 = (l + | A | ) * ( l + | x | ) - " ( l + \x\) Q 2 +a  + a  and let B(a) = (1 + \A\) (l +a  +a  + |a:|)- and C(a) = (1 + | x | ) a  • for a = A + | , C(a) is bounded, since  + Q  l  Q i Then,  decays as —\—  and • for a = z6, i?(i&) is bounded being a product of unitary operators • for a = 1 + ib, we have + |A|)+"(1 + |s|)-*|| = <  ||(1 + \A\y»(l + |A|) (1 + I x D - ^ l + \x\)- \\ 1  ib  ||(1 + |A|)« ||||(1 + L4|) (l + IxD^IHKl + \x\)- \\ 6  :  ib  but ||(1 + | A | ) | | and||(l + |a:|)* || are bounded by unitarity. Hence we need to show l6  6  that ||(1 + |A|) (1 + M ) - ! ! is bounded : Let $ = (1 + Ircl) *. 1  1  n*{2(x -N) + I)HASH  -1  (2{x -N) + i ) n = ( i r - n)(2(x - AO +1) - [x, n], thus  <  | | ( n * - n ) ( 2 ( x - ^ ) + i ) $ | | + ||[x,n]$||  <  C\\(X - N)$\\ + D  <  Recall that A =  C\\X$\\+D  hence ||(1 + lAI) *!! < ||(1 + M ) ^ and ||(1 + |A|) (1 + \x\)- V\\ < C | | * | | =• B(l + ib) 1  1  X  l  is bounded. • Thus B{a) is an analytic family of operators in the strip 0 < Re(a) < 1 that is uniformly bounded on the boundary of the strip. So, by complex interpolation we get that B(a) is bounded everywhere inside the strip. In particular, B(^ + 8)\s bounded. Hence (1 + \A\) Q^ +a  is bounded for a = \ + 6 as well as its adjoint  establishes that indeed Q^RQ^ is bounded.  40  + \A\) . +a  This  Bibliography [AAT] Abou-Chacra R., Anderson P.W., Thouless D.J. A selfconsistent theory of localization J. Phys. CSolid State Phys., Vol. 6, 1973. [Ag] Agmon S. Spectral Properties of Schrodinger Operators and Scattering Theory Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) Vol.2 No.2 151-218, 1975. [AJS] Amrein W.O., Jauch J.M., Sinha K.B. Scattering theory in quantum mechanics. Lecture notes and supplements in physics no 16. 1977. [BD] Ben-Artzi M . , Devinatz A. The limiting absoption principle for partial differential operators Amreican Mathematical Society Vol. 66 No. 364, 1987. [CFKS] Cycon H.L., Froese R.G.,Kirsch W., Simon B. Schrodinger operators. Berlin, Heidelberg, New York : Springer, 1987. [En] Enss V. Asymptotic completeness for quantum mechanical potential scattering I. Shortrange potentials. Comm. Math. Phys. Vol.61 No3:285-291, 1978 Completeness of three body quantum scattering. Lecture Notes in Math., 1031, Springer, Berlin-New York, 1983. [Gr] Graf G.M. Asymptotic Completeness for N-Body Short-Range Quantum System: A New Proof Comm. Math. Phys. 132 nol:73-101, 1990. [HP] Hutson P., Pym J.S. Applications of Functional Analysis and Operator Theory. Mathematics in Science and Engineering, Vol.146, Academic Press 1980. [L]  Lavine R.B. Commutators and Scattering Theory I. Repulsive Interactions Comm. Math. Phys., 20:301-323, 1971. Commutators and Scattering Theory II. A class of One Body Problems Indiana Univ. Math. Jour., Vol.21 No.7:643-656, 1972.  [Mol] Mourre E. Absence of singular continuous spectrum for certain self-adjoint operators. Comm. Math. Phys., 78:391-408, 1981. [Mo2] Mourre E. Operateurs conjugues et proprietes de propagation. Comm. Math. Phys., 91:279-300, 1983. [P6] Poschel Examples of Discrete Schrodinger Operators with Pure Point Spectrum. Comm. Math. Phys. 88:447-463, 1983. [PSS] Perry P., Sigall.M., Simon B. Spectral analysis of N-body Schrodinger operators. Annals of Math. 114:519-567, 1981. [RS3] Reed M . , Simon B. Methods of modern mathematical physics. Vol. III. New York: Academic Press, 1979. [Ro] Royden H.L. Real Analysis 3rd edition. Macmillan Publishing Company, 1988. [S]  Simon B. Some Jacobi Matrices with Decaying Potential and Dense Point Spectrum. Comm. Math. Phys. 87:253-258, 1982. 41  Bibliography [Si]  Sigal, I.M. Scattering theory for many-body quantum mechanical systems. Rigorous results. Lecture Notes in Mathematics, 1011 Berlin-New York, Springer, 1983.  [SSo] Sigal I.M., Soffer A. The N-particle scattering problem: asymptotic completeness for short-range systems Annals of Mathematics 126:35-108, 1987. [We] Weidmann J. The virial theorem and its application to the spectral theory of Schrodinger operators Bull. Am. Math. Soc. 73:452-456, 1967.  42  Appendix A Derivation of A  This appendix sheds some light on the formula for A. Recall A was given by A = -(U*(2(X - N) + 1) - (2(X — N) + 1)IT) = 1(A* - A), l  (A.l)  We can use this equation to express (Acj))(v) in terms of the components of (f>(w) for w in both the preceding and successive spheres, either directly or via the Mk subspace decomposition. Following the subspace decomposition we find that Au can be splintered out as follows: (we omit the factor | without loss of generality) A\M YO  =  (3aio,  5a20 - 3aoo,  7ci!3o - 5aio,  9a4o - 7a o,  A\ ?i  =  (0,  3a i,  5a i - 3 a n ,  7a i - 5a i,  A\M ^2  =  (0,  0,  3a ,  5a42 — 3 a ,  A|A^3^3  =  (0,  0,  3a ,  5a4 - 3 a ,  AI.M4T4  =  (0,  0,  0,  3«44  A\M ^5  =  (0,  0,  0,  3«45  0  Ml  2  5  3  2  3 2  3 3  2  4  2  2 2  3  2 3  where T = ao Qko + ai Qki +ct2kQk2 + -- • G Mk,for{gko, g \, Qk2, •••} elements of the discrete fc  k  k  k  Haar basis, and <j> = X^Yfc- Hence, summing up components we get that  43  Appendix A. Derivation of A  A(f>(v) = | { 3a Qoo(v)  + (5a - 3a o)£io(w)  W  20  + (7OJ -  0  + ^a iQu(v)  5CXIQ)Q2O(V)  30  + (5OJ I - 3a )Q2i{v)  2  3  + 3a  +...  n  (t>)  +...  + 3a e23(t;)  + ...  3 2  £  2 2  +  3 3  • • •  (A.2)  where Qij{v) is the jth component of the zth Haar basis element. Generalizing this for n = \v\ we have that:  P(v) = \ {  A(  [(2n + 3 ) a  - (2n + l ) a _  ( n + 1 ) 0  (n  n-2  2  ,  i + 1  1)0  ] •  Q o(v) + \(2n + n  «(n-l)l] • QnM + J2\ H K ~ & ~ )) 2n  YI  0«i( ) f + 2 - l > j=2»-i u  n  1=1  1  l)a  ( n + 1 ) 1  - (2n - 1)  -l X  a  (n+l)j  —  (2« — (2/ + l))0(„_ )j]1  j=2<  (n+l)jQnj(v)  }  3a  (A.3) Further by grouping terms (A.3) can be rewritten as : n-l  A<t>{v)  =  |{  (  2 n  +  3  ) (n+i)o-ftio(«) a  + (2n +  l)a(„+i)i-ft,i(w)  +  J ^ ( 2 n - (2i - 1))-  f=i 2'+!-l E  a  (n+l)j  -  0"J'( ) V  (  _  2n  + ^"(n-ljO  1  Qno{v)  -  (2n -  l ) a ( „ _ i ) i •£nl(^) "  J=2' 2  n-2  £ ( 2 n - (21 + 1)) 1=1  , + 1  -l  Y,  (n-i)j  a  •  Qnj(v) }  j=2<  (AA) But we also used an expression of (A<p)(v) in terms of not the components, but of u(iy)'s themselves, for w in both of the preceding and successive spheres. We recall this expression here, explain the notation and proceed to show that the two expressions mentioned for A in (A.4) and (A.5) are equivalent. Expanding (3.19) a bit for the sake of the derivation to come, we have:  / .  \ ">es  n+1  i=i  ^ ™es„+i  \'r(v,w)eSo  y(v,w)£Si  44  '  "<es  n+1  •y(v,w)eS  n  Appendix A. Derivation of A (  n-2  vm ,  ,  v  E <&) + E M E  r  w  E  ^H) +  7(i>,u;)eSi  \7(u>M')6So  (A.5)  7(u,«))es„_i  L e t ' s take the t i m e to e x p l a i n the notation a bit more, as well as to put it into picture a n d words, w h i c h makes it more transparent. Referring to a picture of the configuration space, i.e. the b i n a r y tree below i n figure 2, we easily see w h a t is going o n ;  4 parent  Figure 2 the inner sums  ^  i n (A.5) are just p i c k i n g up a l l the values of (f)(w) for w  and  wes  ^es _  n+1  n  •y(v,w)eSi  1  r/(v,w)eSt  i n either i n the preceding w € 5 _i  or successive w G Sn+i  n  spheres, w i t h / referring to the  sphere Si i n w h i c h the youngest c o m m o n ancestor of v a n d w, j(v,w), lies, w h i l e the outer s u m j u s t sums up the contributions emerging from each sphere 5/ b r i n g i n g i n a factor of an>i or b ^. n  Also, a  n%n  a n d & , _ i are the coefficients m u l t i p l y i n g 0(u;)'s where w's are respectively n  n  the c h i l d r e n a n d the parent of v. So for example if v is the most upper node i n S3 as i n figure 2, then 03,3 is the coefficient of 4>{w) for IO'S the children of v, a 2 is the coefficient before a l l (j>(w), IU'S i n 54, where the 3)  youngest c o m m o n ancestor a(v, w) of v a n d w is the first upper node i n S2, 03,1 is the coefficient before a l l 4>(w), t o ' s i n S4, where the youngest c o m m o n ancestor a(v, w) of v a n d w is the second u p p e r node i n S2 a n d 03^ = 1, while 63,2 is the coefficient of 4>{w) for w the parent of v, 45  63,1  Appendix A. Derivation of A is the coefficient before all <j)(w), u;'s in 53, where the youngest common ancestor a(v,w) of v and w is the first node in S\ and 63,0 = 1. Next we show that (A.4) is equivalent to (A.5). We do this by actually deriving (A.5) from (A.4). We can use the discrete Haar basis decomposition of each Z (S +i) and Z (5 _i), to 2  2  n  n  express each of the a( i)j's and a(„_i)j's ,j running from 0 to 2™ — 1, in terms of sums over n+  <£(«;)'s for w G S \  and G S -\.  n+  As before ~/(v,w) corresponds to the youngest common  n  ancestor of v and w andfi(v,w)corresponds here to the oldest ancestor of v and w sitting in Si- With this notation we have:  T^+T) • 5 2 <K«>) =2aa( i)o0no(«); n+  Y  T^iT • I  ^(*°)-  /3(ti,u))eSi  * iV "  {  =25a  52  ( n + 1 )  i0ni(t;);  7(-u,«))eSo  52  ~~  52  =  ^  2  ( {n+\)2Qn2{v) a  + a( ) Q (v)); n+1  3  n3  i{v,w)€Si  /3(u,u;)e52  and in general we have:  • T O T  ^H-  52 /?(«,w)gS/  52  ^H[=2 f 2  i  52 a  (n+1)i  ft,j(«)  ~/(v,w)eSi  + 1  for / = 1... n — 1. The above sums can be considered as projections onto the different subspaces spanned by individual basis vectors, the location of v determining the the values of g j(v) { 1, -1 Or 0 } in n  Similar identities also hold for w G 5 _ i . Specifically we have that:  (n+i)jQnj-  a  n  T^if  2  '  52  =  2 3 a  n^pj ' j 2  V  52 D€S„_  52 wes _! 7(u,w)65  5=r'{ • -  52 ,„c<; . "€S _ N  1  P(v,w)£S  2  ^  f  =  2  ^ (n-i)ift»i("); a  n  1  /?(u,w)eSi  2  (n-l)O0nO;  weSn-l  -  0  52 ^ ) ^ " ^ ( ^ j t e W + a ^ ^ t o f f l ) ; ...cc . J wes _ n  1  -y(v,w)eSi  46  Appendix A. Derivation of A and in general we have: 2'+!-l #  TE ^IS]  E  W)~  E  •>€S _ n  W)\  ^>€S _  i  n  =2^ E  (n-l)jQnj{v)  a  J 1  for I = 1... n - 2. Substituting the appropriate factors into (A.4) we get that: (/ (2n + 3) TSHT • E 2  E *M  E  _, n - l  2~  n+ l  n  /3(v,w)6Si  E  E  iu6S  /3(u,w)€52  E  W) -  7(^,^)e5i  + ...+ (2n-(2f-l)) 2^  s  ~ E w)  52  + ... + 3  +1  { (2n + 1)  2  2  E *(«o  wes„-i  - (2n-l)  • - j ^ a ( 2 2^  E  E *M-E  ±=z(  L2  2  ^)  7(^,i«)e5o  - (2n-5)  S  /3(^,«;)G52  ~y(v,w)€Si  (2n - (21 + 1))  2i.  *M  «<eSn-i  P(v,w)€Si  E  ~ E w) 3  52  (u  ™6 n-l  V  P(v,w)es  0 ,)_ n  1  - (2n - 3) ^  2 ^  P{v,w)es  7(l),U))6S;  0(v,w)€Si  1  ~y{v,w)eS2  3  E  + (2n-3)-  n + 1  »6 n+l  P{v,w)es  E  wes .  2  + (2n-l)  7(«,tu)eSo  E  W) + (2n + l)  1  2  2  2  2  ( E wy  y(v,w)eS2  r  E ()  (n-2)  U W  7(D,U;)6S(  2  1  /  ^~-7f-(  ,  ___  E  E  /3(u,io)eS„_i  7(v,ui)e5„_  x  *(«o 2  / J  (A.6) Further factors of -y=r and  can be taken out and (A.6) can be written in the more general  47  Appendix A. Derivation of A form:  (A4) =  (2n + 3)--- J2 -.  '  1  2 2t 12^  n—1  EP"-^- ))' '" .  ' wes  i=o"  n+1  ( E  \  E  0(v,w)es  %  1  '  (2„ + ! ) . ! .  £  0 ( u  ,)_  u)G5„_i (  n-2 X;(2n-(2/  2  / 1  7(«,tu)GS  l+1  1  J2  + l)).2^-f  \1  - E  u'€S„_  ^(u,w)65 i  U  H J  1  7(u,u;)6S|  /+  (A.7) Time to draw a picture. We wish to show that (A.7) is equivalent to (A.5). To do so we need to show that the coefficient before each <j>(w) is indeed given by the ones in (A.5). The different factors before any (j>(w) in (A.7) can easily be summed up to this effect. Figure 3 below gives a transparent representation of (A.7) in the case |u| = 3 and v is the upper most node as shown. a a b, b  0  n  a,  b  Figure 3 The dark lines indicate to add the corresponding above factor while the dotted lines indicate to subtract the corresponding above factor. The horizontal arrows indicate which factors contribute to (f>(wys coefficient. Specifically we have that  48  Appendix A. Derivation of A  a + a + ai + . . . + a _ 0  = -(2n + 3) +-(2n+ 1) + (2n - 1) + (2n - 3) • 2 + . . . + 3 • 2"~ l  n  x  l  2  n-l  =  n + §+ ^(2n-(2it-l))2 -  =  a „ as in (3.20) of section 3.4.2  f c  n )  i-i  1-1  a + a + J2 k - a k=l a  0  1  = -((2n + 3) + (2n + 1))  t  J ^ ( 2 n - (2/fc - 1))2*  +  - (2n - (2Z - 1))2'"  _1  fc=i l-i  2n + 2 + J^(2n - (2* - l))2 -  =  fe  =  - (2n - (2/ - 1))2'  1  a i as in (3.20) of section 3.4.2 1 = ^(2n + 3) - ]-(2n + 1) = 1 = a 2^ ' 2'  anda-ao  _1  nj  as in (3.20) of section 3.4.2  n>0  «-i  l-i  b + bo + J2°k ~ k = ((2n + l) + {2n-l)) + J2(2n-(2k + l))2 1  k  1  2  fc=i  - (2n - (21 + 1))2'~  :  fc=i l-i  2n + ]T(2n - (2/c + l ) ^ "  =  1  l-i - (2n - (21 + 1))2  fc=i  = 6 + 6 + 6i + . . . + 6 _ 0  n  2  6 / as in (3.21) of section 3.4.2 n>  = i(2n + l) + ^ ( 2 n - l ) + ( 2 n - 3 ) + . . . + 3 - 2 " -  3  n-2  = n + i + ]T(2n - (2/c + l)^*  -1  A;=0  = and 6 - 6  6„ _i in  as in (3.21) of section 3.4.2  = \(2n + 1) - \(2n - 1) = 1 = 6 , as in (3.21) of section 3.4.2 2 "~ • 2 Hence grouping up terms (A.7) can be rewritten as  (A7>  -  0  n  V  E«>/ f E ft™)  0  a  _ n-l  2—  1=0  ^  » € S  n  +  1  i{v ,u»)e5j  n —3  2~  XX* ( E 1= 0  ^ u-eSn.!  which is exactly what is given in (A.5). Hence (A.4) and (A.5) are equivalent.  49  '  Appendix B The double commutator [[Q,iA],iA]  B.l  Definitions  Definition of the conjugate operator A:  (MM  ik = ^5>,« 1=0  E \  u>€S  n-l  -^EM  *w  1=0  n + 1  7 ( f ,w)eSi  E  *w  (B.l)  7(«,u')e5i  Simple commutator:  kn  E ".' ( E ccu,^)^)  ^Dv( E  a  C K ^ M )  1=0  V  1=0  u.es _! n  /  (B.2)  B.2  Derivation of the double commutator  Double commutator: Let B(j)(v) = s(v), A<j)(v) = t{v). Then [B,iA](f>(v) = iBA<f>(v) - iABcf>{v) = iBt(v) - iAs(v) and by grouping grouping terms up in (B.2) we have: (B.3) =  yEv( 1=0  C(«,«,)«(«,))  E V  ™ € S  n  +  1  '  -  ^EV/( 1=0  7(ii,w)eS(  C(«, «,)*(«,)  E ^  » 6 S  n  _ j  7(U,U))GS;  50  (B.3)  Appendix B. The double commutator [[Q,iA],iA]  -yl^ nA  2^  a  1=0  »€S„  V  +  M )  S  o ~  /  1  5>,»(  E «H)  7(«IW)€SJ  7(II,UI)G5J  Using the appropriate expressions of t(w) and s(u>), and letting p = n + 1 and r = n — 1, the expression for [B,iA]4>(v) is rewritten as B.Z = "  / i=0  n  +  1  \  fc=o  <•  » e s •y(v,w)eSi  v  zes  z)  k=0  '  n+2  *es„_  V  •y(w,z)£S  7(w,z)e5j  k  (n-l  +2 7(io,2)es  I /=0  v  *>es -y(v,w)eS,  fc=0  1  Ev/( E {|E^( E  n  /  • /=0  ^ »€S„_  \  ^  X  A;=0  ^  7(o,w)eS;  *€S  '  n + 2  7(w,2)€5  fc  /  1  6r fc  fc=0  k=0  ( _ im 2  E '( E ~2 _  '  k-  k  C K ^ W  r  *es r/(w,z)eS n + 2  \  7(W,2)65A;  m.rI  <-  n+1  V  E 7(^,2)65*.  f c  E {yXX*( E a-,W)-^XX  EM E  ;  r-l  X X ( E c K - ) { f XX* ( E *( *)) - ^ E M  —I  2  z€S„_  v  C(«>,*)0(z) 2  7(u),«)e5  fc  Pulling in the potential terms and letting C,(v,w,z) = 2q(w) - q(v) - q(z), the above can be rewritten on the next page as [B,iA]<f>(v) — ^ = 2  n  ,  EM  . 1=0  ,  p  ,  .  p-1  .  E {fE<W E c(wW)- ^ E ^ 6  v  «<€€SS  n  + 1 n + 1  ^  k=0  ^  '  * e st+2 n + 2  fc=0  -y(w,z)€S  k  E c(ww*))}) - ^ ( x x ( E { ^ X X ^ E c( W) V V W  *6 n-2 7(w,2)eSjt S  1=0  V  -"eS„_!  j(v,w)eSi  -^(XX( E fe=0  C(  W  v  fc=0  V  *€S  n + 2  7(iu «)eSj. )  )^))})  z6S _2 n  7(tu,2)e5  fc  Having this at hand we can use the same trick as in Lemma 3.1, letting \2q(w) — q(z) - q(w)\ 51  Appendix B. The double commutator [[Q,iA], iA] decay faster than  Let D = P [B,iA], Then summing up its matrix elements, which can n  be picked out of the above formula, using the formulas (3.22) and (3.23) for ci(-) and c {-), so 2  as to use Schur's lemma, we get that : supV|Dgl  "  <  ci(n)(ci(p) + C2(p)) + C2(n)( (r) + C2(r)) = Cl  w i(9n + 24n + f ) = A n + A n + A = M (n) = 0(n ) 2  2  2  x  2  3  3  and similarly s u p ^ \ J J\ ^ M {n) = 0{n ). Hence w —' ' D  2  n  3  V  (J  I\D\| < M ( n ) ^ — = 0 ( n ) _£  3  52  which -» 0 as n -» oo.  

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