Limiting absorption principle for discrete Schrödinger operators with a Wigner-von Neumann potential and a slowly decaying potential Sylvain Golenia, Marc-Adrien Mandich

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Sylvain Golenia, Marc-Adrien Mandich. Limiting absorption principle for discrete Schrödinger opera- tors with a Wigner-von Neumann potential and a slowly decaying potential. Annales Henri Poincaré, Springer Verlag, 2021, 22 (1), pp.83-120. ￿10.1007/s00023-020-00971-9￿. ￿hal-02474104v2￿

HAL Id: hal-02474104 https://hal.archives-ouvertes.fr/hal-02474104v2 Submitted on 22 Jan 2021

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. LIMITING ABSORPTION PRINCIPLE FOR DISCRETE SCHRÖDINGER OPERATORS WITH A WIGNER-VON NEUMANN POTENTIAL AND A SLOWLY DECAYING POTENTIAL

SYLVAIN GOLÉNIA AND MARC-ADRIEN MANDICH

Abstract. We consider discrete Schrödinger operators on Zd for which the perturbation con- sists of the sum of a long range type potential and a Wigner-von Neumann type potential. Still working in a framework of weighted Mourre theory, we improve the limiting absorption principle (LAP) that was obtained in [Ma1]. To our knowledge, this is a new result even in the one-dimensional case. The improvement is twofold. It weakens the assumptions on the long range potential and provides better LAP weights. Both upgrades include logarithmic terms. The proof relies on the fact that some particular functions, that contain logarithmic terms, are operator monotone. This fact is proved using Loewner’s theorem and Nevanlinna functions.

1. Introduction The limiting absorption principle (LAP) is an important resolvent estimate in the spectral and scattering theory of quantum mechanical Hamiltonians, in particular it implies that the part of the continuous spectrum where it holds is actually purely absolutely continuous. The LAP also provides boundary values for the resolvent that are useful for scattering theory. For Schrödinger operators on Rd, the LAP was derived for a large class of short and long range potentials, see e.g. [A], [PSS], [ABG], as well as for decaying oscillatory potentials such as the Wigner-von Neumann potential, see e.g. [DMR], [RT1], [RT2]. Schrödinger operators with Wigner-von Neumann potentials [NW] are of interest because when appropriately calibrated they may produce eigenvalues embedded in the absolutely continuous spectrum of the Hamiltonian, and are linked with the phenomenon of resonance. The Wigner-von Neumann type potentials are still very much an active area of research. For the multi-dimensional case, see e.g. [FS], [GJ2], [J], [JM], [Mar1], [Mar2], [Mar3] and [Ma1]; for the one-dimensional case, see e.g. [Li2], [L1], [L2], [L3], [Sim], [JS], [KN], [NS] and [KS]. Regarding the LAP for Schrödinger operators, historically a lot of effort was put into lessening the decay assumptions on the short/long range perturbations. In this regard a breakthrough was made in 1981 by E. Mourre, see [Mo1] and [Mo2]. Very roughly speaking, assuming the ´2 short range perturbation satisfies Vsr x O x and the long range perturbation satisfies ´1 p q “ p| | q x ∇Vlr x O x as x , as well as several other technicalities, Mourre’s theory ¨ p q “ p| | q | | Ñ `8 d implied the following LAP for the Schrödinger operator H ∆ Vsr Vlr on R over some open real interval I: for every fixed s 1 2, “ ´ ` ` ą { (1.1) sup A ´s H z ´1 A ´s , I˘ : z C : Re z I, Im z 0 . zPI˘ }x y p ´ q x y }ă8 “ t P p q P ˘ p qą u Here x : 1 x 2 and A is some self-adjoint operator. The theory was improved and by x y “ ` | | the end of the 20th century the book [ABG] presented a more sophisticated but optimal abstract a framework to treat the short/long range potentials. This framework now goes by the name of

2010 Mathematics Subject Classification. 39A70, 81Q10, 47B25, 47A10. Key words and phrases. limiting absorption principle, discrete Schrödinger operator, Wigner-von Neumann potential, Mourre theory, weighted Mourre theory, Loewner’s theorem, polylogarithms. 1 LAPFORDISCRETESCHRÖDINGEROPERATOR 2 classical Mourre theory. The Wigner-von Neumann potential is not in the scope of this framework because its decay is at best O x ´1 , see e.g. [GJ2, Proposition 5.4] for the continuous case and [Ma1, Proposition 4.2] for thep| discrete| q case. The beginning of the 21st century sees the emergence of new approaches to Mourre’s commu- tator method with [G] and [GJ1], and the so-called weighted Mourre theory, see [GJ2], or the local Putnam-Lavine theory, see [J]. The classical theory revolves around the Mourre estimate

(1.2) EI H H, iA EI H γEI H K, p qr s˝ p qě p q` where EI H is the spectral projection of H onto the interval I, H, iA ˝ is the realization of the formal commutatorp q between H and iA, K is a compact operator,r and γs 0. On the other hand the more recent theories are centered around an estimate of the type ą ´s ´s (1.3) EI H H, iϕ A EI H EI H A γ K A EI H , s 1 2, p qr p qs˝ p qě p qx y p ` qx y p q ą { where ϕ is the real-valued function t (1.4) ϕ t : x ´2sdx, t R. p q “ x y P ż´8 Key underlying ideas behind this choice are that 1) ϕ is bounded for s 1 2 and, 2) ϕ1 A A ´2s, i.e. the derivative of ϕ yields the appropriate weights for the LAPą as{ in (1.1). p q “ x Iny [GJ2] ideas of weighted Mourre theory are used to derive the LAP for the continuous d Schrödinger operator H ∆ Vsr Vlr W on R , where W x w sin k x x , w, k R, is the Wigner-von Neumann“ ´ potential,` ` and` the short/long rangep potentialsq “ p respectively| |q{| | satisfyP V x O x ´1´ε and x ∇V x O x ´ε as x for some ε 0. The cost of srp q “ p| | q ¨ lrp q “ p| | q | | Ñ `8 ą including the W was that the decay assumptions for Vsr and Vlr are suboptimal as per [ABG, Chapter 7]. Article [Ma1] was an application of the ideas of [GJ2] to corresponding (albeit non-radial) discrete Schrödinger operators on Zd, see Theorem 1.6 below for a statement. This article is a sequel to [Ma1]. The aim is to follow the weighted Mourre theory as in [GJ2] and [Ma1], but with a new bounded function ϕ in (1.3) which involves logarithmic terms. To write down this function we need some notation. Denote ln the Napierian logarithm. Let p¨q ln0 x : 1, ln1 x : ln 1 x , and for integer k 2, lnk x : ln 1 lnk´1 x . Thus k is thep numberq “ of timesp q “ the functionp ` q ln 1 x is composedě withp itself.q “ Forp simplicity` p weqq choose the p ` q p p domains lnk : 1, . Also denote ln x : lnk x , p R. We choose ϕ to be r `8q Ñ `8 kp q “ p p qq P t m (1.5) ϕ t : x ´1 ln´2p x ln´1 x dx, t R, p q “ x y m`1px yq k px yq P ´8 k“0 ż ź for some m N and p 1 2. Note that ϕ as in (1.5) is an increasing function that is asymptot- P ą { 1 2 ically equal to a constant minus a O ln ´ p t term for t , hence a bounded function. p m`1 p qq Ñ `8 More notation is needed to present our results. The position space is the lattice Zd for integer Zd 2 2 2 H 2 Zd d 1. For n n1, ..., nd , set n : n1 ... nd. Consider the Hilbert space : ℓ . ě “ p q P | | “ ` th` “ p q Let Si be the shift operator to the right on the i coordinate : d Siψ n : ψ n1, ..., ni 1, ..., nd , n Z , ψ ψ n Zd H, and 1 i d. p qp q “ p ´ q @ P “ p p qqnP P ď ď The shifts to the left on the ith coordinate are S˚ S´1. The discrete Schrödinger operator i “ i (1.6) H : ∆ V W “ ` ` acts on H, where ∆ is the discrete Laplacian operator defined by d d ˚ (1.7) ∆ : 2 Si Si ∆i, “ i“1 ´ ´ “ i“1 ÿ ÿ LAPFORDISCRETESCHRÖDINGEROPERATOR 3

W is a Wigner-von Neumann potential, parametrized by w R, k 0, 2π π and defined by P P p qzt u ´1 d (1.8) W ψ n : w sin k n1 ... nd n ψ n , for all n Z ,n 0, and ψ H, p qp q “ ¨ p p ` ` qq| | p q P ‰ P W ψ 0 : 0, and V is a multiplication operator by a bounded real-valued sequence V n Zd p qp q “ p p qqnP such that V ψ n : V n ψ n . We will also consider the oscillating potential W2 given by p qp q “ p q p q n1`...`n ´1 d (1.9) W2ψ n : w 1 d n ψ n , for all n Z ,n 0, and W2ψ 0 : 0. p qp q “ ¨ p´ q ¨ | | p q P ‰ p qp q “ ˚ In this case let H2 : ∆ V W2. Denote τiV , τ V the operators of multiplication acting by “ ` ` i ˚ τiV ψ n : V n1, ..., ni 1, ...nd ψ n , τ V ψ n : V n1, ..., ni 1, ...nd ψ n . p q p q “ p ´ q p q p i q p q “ p ` q p q Hypothesis (H) : Assume there are m N and 0 r q such that : P ď ă m (1.10) V n O ln´q n ln´r n , as n , and p q“ m`1p| |q k p| |q | |Ñ`8 ˜ k“0 ¸ źm ´q ´r (1.11) ni V τiV n O ln n ln n , as n , 1 i d. p ´ qp q“ m`1p| |q k p| |q | |Ñ`8 @ ď ď ˜ k“0 ¸ ź Remark 1.1. If hypothesis (H) holds for some m N and 0 r q, then it holds for any integer m1 m and 0 r1 q1, with r1 r and q1 Pq. ď ă ě ď ă ď ď Under hypothesis (H), V n o 1 as n . Thus V W is a compact perturbation p q “ p q | | Ñ `8 ` of ∆ and so σess H σ ∆ . A Fourier transformation calculation shows that the spectrum of ∆ is purely absolutelyp q “ continuousp q and σ ∆ 0, 4d . The formulation of the LAP requires a p q “ r s th conjugate self-adjoint operator. Let Ni be the position operator on the i coordinate defined by

2 d 2 Niψ n : niψ n , Dom Ni : ψ ℓ Z : niψ n . p qp q “ p q r s “ P p q Zd | p q| ă8 " nÿP * The conjugate operator to H is the generator of dilations denoted A and it is the closure of d i d (1.12) A : i 2´1 S˚ S S˚ S N S S˚ N N S S˚ 0 i i i i i 2 i i i i i i “ i“1 p ` q ´ p ´ q “ i“1p ´ q ` p ´ q ÿ ÿ with domain the compactly supported sequences. A is self-adjoint, see e.g. [GGo]. Let

µ H : 0, 4 E˘ k for d 1, (1.13) p q “ p qzt p qu “ µ H : 0, E k 4d E k , 4d for d 2. p q “ p p qq Y p ´ p q q ě Here k is the angular frequency of W , E k : 2 2 cos k 2 , E k : 2E k for k 0,π ˘p q “ ˘ p { q p q “ ´p q P p q and E k : 2E` k for k π, 2π . The sets µ H are real numbers where a classical Mourre estimatep q (1.2“ ) is knownp q toP hold p forq H – assumingp theq presence of the oscillatory perturbation, i.e. W 0. The first basic result for H is about the local finiteness of the point spectrum. ‰ Proposition 1.1. Let d 1, and V satisfy hypothesis (H) for some m N and 0 r q. If E µ H , then there is aně open interval I of E that contains at most finitelyP many eigenvaluesď ă of H P(includingp q multiplicities). The corresponding eigenfunctions, if any, decay sub-exponentially. Also, in dimension 1 the point spectrum of H is empty in I whenever I E k , E k . Ă p ´p q `p qq We have a similar result for H2. Set µ H2 : 0, 4d 0, 4, ..., 4d 4, 4d 2d . p q “ r sz pt ´ u Y t uq Proposition 1.2. Let d 1 and V satisfy hypothesis (H) for some m N and 0 r q. If ě P ď ă E µ H2 , then there is an open interval I of E that contains at most finitely many eigenvalues of P p q H2 (including multiplicities). The corresponding eigenfunctions, if any, decay sub-exponentially. In dimension 1 the point spectrum of H2 is void in 0, 4 2 . p qzt u LAPFORDISCRETESCHRÖDINGEROPERATOR 4

Denote P K : 1 P , where P is the projection onto the pure point spectral subspace of H. We introduce the“ LAP´ weights with logarithmic terms : M 1 1 ´p ´p,´ 2 α,β β (1.14) W x : x ´ 2 w x , w x : lnα x ln x , α, β R. M`1p q “x y M p q M p q “ M`1 px yq k px yq P k“0 ź Let J ˘ : z C : Re z J, Im z 0 . The main result of this article is : “ t P p q P ˘ p qą u Theorem 1.3. Let d 1 and V satisfy hypothesis (H) for some m N and 2 r q. If E µ H , then there isě an open interval I of E such that for any compactP interval“J ăI, any integerP p Mq m, and any p 1 2, Ă ě ą { W´p ´1 KW´p (1.15) sup M`1 A H z P M`1 A . zPJ˘ p q p ´ q p q ă8 › › The following local decay estimate› also holds for all ψ H, M ›m and p 1 2 : › P ě › ą { 2 W´p ´itH K (1.16) M`1 A e P EJ H ψ dt . R p q p q ă8 ż › › › › 2 2 1{2 By Lemma 4.11 the above estimates› also hold for N ›1 N1 ... Nd instead of A . Finally, the spectrum of H is purely absolutelyx continuousy “ p ` | on|J `whenever` | P| q 0 on J. x y “ With the obvious substitutions, the statement and conclusion of Theorem 1.3 also hold for the Hamiltonian H2 ∆ V W2. “ ` ` Remark 1.2. If W 0 or W2 0, then the conclusion of Theorem 1.3, i.e. the existence of a neighborhood where“ the LAP“ holds, is valid for a larger set of energies, precisely E 0, 4d 0, 4, ..., 4d 4, 4d . P r szt ´ u Example 1.4. Suppose V n O n ´1 ln´q n as n for some q 2. Then hypothesis p q“ | | | | | |Ñ`8 ą (H) holds with m 0 and r 2. The LAP in Theorem 1.3 holds with the weights W´p A ` ˘ M`1 “ “ ´p 1 p q for any M 0 and p 1 2, so in particular for W A A ´ 2 ln´p 1 A , p 1 2. ě ą { 1 p q“x y p `x yq ą { Example 1.5. An interesting 1-dimensional example is H2 ∆ V W2 where V n n ´1 n ´1 “ ` ` p q “ 2 1 n ln n and W2 n 1 n , n N. It is due to C. Remling [R]. Although the examplep´ q ¨ pis onp theqq half-line thep q“p´ argumentsq ¨ presentedP here apply with minor considerations. So in particular the spectrum of this H2 is purely a.c. on 2, 2 0 (in the notation of [R]). p´ qzt u For comparison, the statement of the principal result of [Ma1] is added below, taking into account the technical improvement in [Ma2, Theorem 1.8] : ´ε ´ε Theorem 1.6. [Ma1] Let d 1 and V satisfy V n O n and ni V τiV n O n for some ε 0, and 1 i ěd. If E µ H , thenp q“ therep| exists| q an openp interval´ qpI containingq“ p| | Eq such that forą any compact@ ď intervalď J PI andp q any s 1 2, sup A ´s H z ´1P K A ´s . Ă ą { zPJ˘ }x y p ´ q x y }ă8 Thus, Theorem 1.3 improves Theorem 1.6 by weakening the decay assumptions on V , as well as improving the LAP weights. The article [Ma1] considers a second Wigner-von Neumann 1 d 1 potential, namely W n i“1 sin kini ni. Although W could have been included in this article and similar resultsp q “ would havep beenq{ derived, we did not do so to keep the size of the article reasonable. Anotherś obvious comment is that we do not know how to effectively use commutator methods to study spectral properties of the discrete Hamiltonian H (with W 0) on 0, 4d µ H , see the comments that follow [Ma1, Proposition 4.5]. For the continuous‰ Schrödingerr sz operatorsp q on Rd, with W 0, a similar limitation exists, see [GJ2] and [JM], in that the LAP was established only on ‰0, k2 4 for d 2. But interestingly, an argument was p { q ě LAPFORDISCRETESCHRÖDINGEROPERATOR 5 recently found to justify a LAP on 0, k2 4 when the Wigner-von Neumann potential is radial, i.e. W x q sin k x x , seep [`8qztJ] and [Mb{ ,u Section 3.5]. Fix d 1 top discussq“ Theoremp | |q{| |1.3 in relation to other one-dimensional results in the literature. In [Sim],“ the perturbation consists of a Wigner-von Neumann potential plus a V ℓ1 Z , and it P p q is proved that the spectrum of H is purely absolutely continuous on 0, 4 E˘ k . Note that our assumption (1.11), with 2 r q, implies V τV ℓ1 Z butp notqzt necessarilyp qu V ℓ1 Z , (although the weaker assumption“ 1ă r q isp sufficient´ q P to havep q V τV ℓ1 Z ). AnotherP p q recent result is [Li1], where it is shown“ thată if the perturbation isp O´n ´1q, P thenp qH does not have singular continuous spectrum. Although this criteria applies to thep| | Wigner-vonq Neumann potential, it does not apply to all potentials V in the scope of this article, since for example hypothesis (H) allows for V n ln´q 1 n , q 2. Now suppose the absence of oscillations, p q“ p `x 1yq ą i.e. W 0 and W2 0. Because V τV ℓ Z , V is of bounded variation, and so the spectra “ “ p ´ q P p q of H and H2 are purely a.c. on 0, 4 , by a result due to B. Simon in [Si]. Consider now d 1 to discussp Theoremq 1.3 in relation to the framework of Mourre theory as exposed in [ABG].ě First, it is important to cite [BSa], where the details of this framework are worked out for the multi-dimensional discrete Schrödinger operators on Zd. From this perspec- tive, this article is an attempt to bridge the gap between [Ma1] and [BSa]. 1,1 1 1 Recall the regularity classes C A Cu A C A that structure the classical Mourre theory, see Section 2.1. In[ABG, Chapterp q Ă 7],p it isq shownĂ p thatq C1,1 A is an optimal class in that if the Hamiltonian H C1,1 A then a LAP holds for H, whereasp q there is an example where 1 1,P1 p q H Cu A but H C A and no LAP holds. In [Ma1, Proposition 4.2] it is proved that the P p q R p q 1 Wigner von-Neumann potential W , and thereby H, are not even of class Cu A . An analogous 1 p q argument also gives W2,H2 Cu A , provided w 0. Thus our Hamiltonian does not fall under the framework of classical MourreR p theory.q ‰ Let us discuss the assumptions on the short/long range perturbations Vsr, Vlr. The criteria in the classical theory (see [BSa, Theorem 2.1], also [ABG, Theorem 7.6.8]) are : 8 (1.17) sup V n dκ , 1 i d, | srp q| ă8 @ ď ď ż1 κă|n|ă2κ and 8 (1.18) V n o 1 as n and sup V τiV n dκ , 1 i d. lrp q“ p q | |Ñ`8 |p lr ´ lrqp q| ă8 @ ď ď ż1 κă|n|ă2κ Table 1 contrasts these criteria with our hypothesis in Theorem 1.3. In this Table g n : ´1 ´q,´r p q “ O n wm n as n . p| | p qq | |Ñ`8

d lnp2`|ni|q If V satsifies V n O g n V τiV n O g n , i V n σ p q“ p p qq p ´ qp q“ p p qq @ p q“ i“1 ln p1`xnyq then : (1.10) holds yes, m N no yes, σ 3 @ P ř@ ą with 2 “ r ă q then : (1.11) holds yes, m N yes, m N yes, σ 2 @ P @ P @ ą with 2 “ r ă q then : (1.17) holds yes, m N no no, d 1 @ P @ ľ and 1 r q “ ă then : (1.18) holds yes, m N yes, m N yes, for d 1, σ 1; @ P @ P “ ą and 1 r q and 1 r q no, d 2 “ ă “ ă @ ľ Table 1. Examples. LAPFORDISCRETESCHRÖDINGEROPERATOR 6

The last column in Table 1 shows that it is possible to concoct a non-radial potential V that verifies the hypothesis of Theorem 1.3 but neither (1.17) nor (1.18). It is an open problem for us to verify if this potential is of class C1,1 A . If the potential V is radial the first two lines of Table 1 suggest that our assumptions onp Vq may be slightly suboptimal, at least in the absence of oscillations, i.e. W 0, because Theorem 1.3 requires r 2, rather than r 1. To the best of our understanding,“ our requirement of "an entire extra“ logarithm" (r 1“vs. r 2) is due to a technical limitation in the construction of the almost analytical extension“ of the“ function ϕ, see (1.5), which is employed to apply the Helffer-Sjöstrand formula and associ- 1 ´1 ´2p,´1 ated functional calculus. Indeed, although ϕ t t wm t the extension only verifies ϕ˜ z c Re z ´1´ℓ Im z ℓ,ℓ N. Clearlyp q“x all logarithmicy decayp q is lost in the process of |Bextending{B | ď tox thep complexqy | plane.p q| P Let us briefly discuss the LAP weights. Let T be a self-adjoint operator on H, EΣ T its 1 j´1 p q j spectral projection on a set Σ R. Let s,p,p R, M N. Let Σj x R : 2 x 2 , Ă P P “ t P ď | |ď u j 1, and Σ0 : x R : x 1 . Define the Banach spaces ě “ t P | |ď u 2 s p,p1 L 1 T : ψ H : T w T ψ , and s,p,p ,M p q “ P }x y M p q }ă8 ! 8 ) H ? j B T : ψ : 2 EΣj T ψ . p q “ P j“0 } p q }ă8 ! ÿ ) The norms on these spaces are respectively 8 s p,p1 2 ? j ψ L pT q : T w T ψ and ψ BpT q : 2 EΣj T ψ . s,p,p1,M M } } “ }x y p q } } } “ j“0 } p q } ÿ 2 2 ˚ 2 The dual of L 1 T with respect to the inner product on H is L 1 T L 1 T s,p,p ,M p q p s,p,p ,M p qq “ ´s,´p,´p ,M p q and the dual B˚ T of B T is the Banach space obtained by completing H in the norm p q p q ˚ ? ´j ψ B pT q : sup 2 EΣj T ψ . } } “ jPN } p q } We refer to [JP] and the references therein for these definitions. Write L2 T : L2 T . For sp q “ s,0,0,0p q any s,p 1 2 and M N, the following strict inclusions hold : ą { P L2 T L2 T B T L2 T , sp qĹ 1{2,p,1{2,M p qĹ p qĹ 1{2p q and L2 T B˚ T L2 T L2 T . ´1{2p qĹ p qĹ ´1{2,´p,´1{2,M p qĹ ´sp q 2 2 1 1 Also clear is that L T L 1 1 T whenever p p and M M. 1{2,p,1{2,M p qĂ 1{2,p ,1{2,M p q ď ě Let E H be the set of eigenvalues and thresholds of H. By threshold we mean a real number E for whichp q no Mourre estimate holds for H wrt. A, regardless of the interval I, I E. Classical Mourre theory says that under appropriate conditions for V , and W 0, then theQ LAP “ ´1 (1.19) sup H λ iη ψ K˚ c λ ψ K ηą0 }p ´ ´ q } ď p q} } holds for some appropriate pair of Banach spaces K, K˚ , some c λ 0 and all λ R E H . Also, c λ can be chosen uniform in λ over fixed compactp subsetsq of Rp EqH ą . On the oneP hand,z p theq spaces p Kq, K˚ B A ,B˚ A are optimal in a certain sense, see [zJPp], [qAH] and the discussion at the beginningp q “ ofp [pABGq , Chapterp qq 7], but these are not Besov spaces. On the other hand, the Besov spaces K, K˚ H , H which appear in [ABG] and [BSa] are not as optimal p q “ p 1{2,1 ´1{2,8q as B A ,B˚ A , but are optimal in the scale of Besov spaces and allow for a larger class of potentialsp p q V .p Noteqq that our new result Theorem 1.3 implies (1.19) for K L2 A , any “ 1{2,p,1{2,M p q p 1 2, M m, while Theorem 1.6 implies (1.19) for K L2 A , any s 1 2. ą { ě “ sp q ą { LAPFORDISCRETESCHRÖDINGEROPERATOR 7

Finally, a few comments about the proof of Theorem 1.3. In essence it is the same as in [GJ2], or [Ma1]. The difference lies in the function ϕ given by (1.5) rather than (1.4). But in order to formulate meaningful conditions on V (hypothesis (H)) this translates into a problem of bounding functions of the generator of dilations A by functions of the operator of position N involving logarithmic terms. More specifically, while the proof of Theorem 1.6 required |only| A ε N ´ε B H , the space of bounded operators on H, the proof of Theorem 1.3 x y x y P p q requires wα,β A w´α,´β N B H for some appropriate α, β 0, which is equivalent to M p q ¨ M p q P p q ě w2α,2β A w´2α,´2β N in the sense of forms. To achieve this we make a detour via the M p q ď M p q polylogarithm functions Liσ z (specifically the ones of order 2 σ 3 are used). The reason for doing this is that higher powersp q of the logarithm are not Nevanlinnaă ď functions, but the functions Liσ z , Re σ 0, are Nevanlinna functions which continue analytically across 1, from p q p q ą p´ `8q C` to C´, and are thus operator monotone functions, see Section 8. The plan of the article is as follows. In Section 2, we recall definitions and results of classi- cal and weighted Mourre theory. In Section 3 we derive the classical Mourre estimate for the Hamiltonians H and H2 and prove Propositions 1.1 and 1.2. In Section 4 we derive operator norm bounds involving logarithms of self-adjoint operators. In Section 5 we prove key Lemmas that are needed for the proof of Theorem 1.3. In Section 6 we prove Theorem 1.3. In appen- dix A (Section 7) we review Loewner’s theorem which makes the connection between Nevanlinna functions and operator monotone functions, and prove an extension of Loewner’s theorem that is suitable for semi-bounded self-adjoint operators. In appendix B (Section 8) we briefly review polylogarithms and explain that the ones of positive order are Nevanlinna functions. In appen- dix C (Section 9) we mention the basic tools we use from the kit of almost analytic extensions and Helffer-Sjöstrand functional calculus. Acknowledgements : It is a pleasure to thank Thierry Jecko for fruitful conversations on the topic, generous advice to improve the results in various ways, and especially giving us the permission to publish his Proposition 4.4 which conveniently generalizes self-adjoint operator norm estimates to sub-multiplicative functions. We are grateful to the two anonymous referees for a careful reading of the manuscript leading to many improvements, and especially Proposition 4.4.

2. Basics of the abstract classical and weighted Mourre theories 2.1. Operator Regularity. We consider two self-adjoint operators T and A acting in some complex Hilbert space H, and for the purpose of this brief overview T will be bounded. Given k N, we say that T is of class Ck A , and write T Ck A , if the map P p q P p q (2.1) R t eitAT e´itA B H Q ÞÑ P p q has the usual Ck R regularity with B H endowed with the strong operator topology. The form T, A is definedp onq Dom A Dom pA qby ψ, T, A φ : T ψ, Aφ Aψ, T φ . Recall the rfollowings convenient result:r sˆ r s x r s y “ x y´x y Proposition 2.1. [ABG, Lemma 6.2.9] Let T B H . The following are equivalent: P p q (1) T C1 A . (2) TheP formp q T, A extends to a bounded form on H H defining a bounded operator denoted 1 r s ˆ by adA T : T, A ˝. (3) T preservesp q “ Dom r As and the operator T A AT , defined on Dom A , extends to a bounded operator. r s ´ r s Consequently, T Ck A if and only if the iterated commutators adp T : adp´1 T , A P p q Ap q “ r A p q s˝ are bounded for 1 p k. We say that T Ck A if the map (2.1) has the Ck R regularity ď ď P up q p q LAPFORDISCRETESCHRÖDINGEROPERATOR 8 with B H endowed with the norm operator topology. We say that T C1,1 A if p q P p q 1 T, eitA , eitA t´2dt . }r s˝ s˝} ă8 ż0 It turns out that C2 A C1,1 A C1 A C1 A . p qĂ p qĂ up qĂ p q 2.2. The Mourre estimate. Let I be an open interval and assume T C1 A . We say that the Mourre estimate holds for T on I if there is γ 0 and a compact operatorP p Kq such that ą (2.2) EI T T, iA EI T γEI T K p qr s˝ p qě p q` in the form sense on Dom A Dom A . We say that the strict Mourre estimate holds for T on I if (2.2) holds with Kr sˆ0. Assumingr s the estimate holds over I, T has at most finitely many eigenvalues in I, and they“ are of finite multiplicity, while if the strict estimate holds T has no eigenvalues in I. This is a direct consequence of the Virial Theorem, see [ABG, Proposition 7.2.10]. Let I E; ε be the open interval of radius ε 0 centered at E R. One considers the p q ą P function ̺A : R R: T ÞÑ ̺A : E sup γ R : ε 0 such that E T T, iA E T γ E T . T ÞÑ P D ą IpE;εqp qr s˝ IpE;εqp qě ¨ IpE;εqp q A A It is known for example that ̺T is lower semicontinuous and ̺T E if and only if E (σ T . For more properties of this function, see [ABG, chapter 7]. p qă8 P p q 2.3. The weighted Mourre estimate & LAP. Let T be self-adjoint in a Hilbert space H. Denote P K : 1 P , where P is the projection onto the pure point spectral subspace of T . In [GJ2] it is explained“ ´ in a more general setting that a projected weighted estimate of the form K K K 2 K (2.3) P EI T T, iB EI T P cP EI T C EI T P , p qr s˝ p q ě p q p q where c 0, B,C are any linear operators satisfying CBC´1 B H , and C self-adjoint and injective,ą will imply a LAP of the form P p q (2.4) sup C T z ´1P KC . zPI˘ } p ´ q }ă8 The proof is by contradiction and therefore does not provide a description of the continuity properties of the LAP resolvent. Nonetheless, the resolvent estimate (2.4) implies the absence of s.c. spectrum for T in I, and if B is T -bounded then (2.3) also gives the time decay estimate

itT K 2 (2.5) Ce EI T P ψ dt , ψ H. R } p q } ă8 P ż 3. Verifying the classical Mourre estimate In this Section we derive the Mourre estimate described in the previous Section for T H given by (1.6). But first we must discuss operator regularity. “ 3.1. Verifying the operator regularity. We use freely Proposition 2.1. Let

(3.1) Θ ∆ : t0, t1, ..., td , where tk 4k, k 0, 1, ..., d. p q “ t u “ “ Recall that our choice of conjugate operator A is the closure of (1.12). This choice is justified by the fact that the commutator between ∆ and A is : d (3.2) ∆, iA ∆k 4 ∆k . r s˝ “ p ´ q k“1 ÿ 1 In particular ∆ C A , and since σ ∆k 0, 4 , k 1, ..., d, the strict Mourre estimate holds for ∆ and A overP anyp q interval I pσ ∆q “Θ r ∆s @ 0“, 4d Θ ∆ . For this reason, we refer to Ť p qz p q “ r sz p q LAPFORDISCRETESCHRÖDINGEROPERATOR 9

Θ ∆ as the thresholds of ∆ (they are also thresholds for H and H2). In fact, one can be more p q specific and prove that if E σk ∆ : tk 1, tk , for some k 1, ..., d, then P p q “ r ´ s “ A ̺ E E tk 1 E tk . ∆p q “´p ´ ´ qp ´ q This can be proved by induction on d and using the nifty result [ABG, Theorem 8.3.6]. One can further show that for all k 1, ..., d, ∆, iA decomposes into the sum of a non-negative “ r s˝ operator and a non-negative remainder bk ∆ , namely p q ∆, iA ∆ tk 1 ∆ tk bk ∆ , r s˝ “ ´p ´ ´ qp ´ q` p q where bk ∆ : 8 k 1 ∆ 16k k 1 ∆i∆j. p q “´ p ´ q ` p ´ q` 1ďi,jďd iÿ‰j A somewhat surprising direct calculation based on functional calculus may show that no strict positivity can be extracted from bk ∆ by localizing in energy. Finally, it is not hard to prove that ∆ C2 A , or C8 A for that matter.p q Moving on to the commutator between the potential V and AP, wep have:q p q d ´1 ´1 ˚ ˚ (3.3) V, iA 2 Ni V τiV Si 2 Ni V τ V S . r s˝ “ p ´ qp ´ q ` p ` qp ´ i q i i“1 ÿ It is readily seen that the assumption (1.11), for some m N, 0 r q, implies that V, iA ˝ is a compact operator. In particular V C1 A . Regarding theP commutatorď ă between the Wigner-vonr s P p q Neumann potential W and A, we have W, iA KW BW where r s˝ “ ` d d ´1 ˚ ´1 ˚ (3.4) KW : 2 W S Si 2 S Si W, “ p i ` q` p i ` q i“1 i“1 ÿ ÿ d d ˜ ˚ ˚ ˜ (3.5) BW : UiW Si Si Si Si WUi, “ i“1 p ´ q´ i“1p ´ q ÿ ÿ ˜ ˜ W is the operator W ψ n : w sin k n1 ... nd ψ n and Ui is the operator Uiψ n : ´1 p qp q “ ¨ p p ` ` qq p q p qp q “ ni n ψ n , for n 0, Uiψ 0 : 0. Note that KW is compact and BW is bounded. Thus W| |C1 Ap .q However,‰ W p C1qpA q, see“ [Ma1, Proposition 4.2]. Finally, regarding the commutator P p q R up q between the oscillating potential W2 and A, we have W2, iA KW2 BW2 where r s˝ “ ` d d ´1 ˚ ´1 ˚ (3.6) KW2 : 2 W2 Si Si 2 Si Si W2, “ i“1p ` q` i“1p ` q ÿ ÿ d d ˚ ˚ (3.7) BW2 : Uiσ S Si S Si σUi, “ p i ´ q´ p i ´ q i“1 i“1 ÿ ÿ n1`...`nd σ is the operator σψ n : w 1 ψ n . Again, KW2 is compact and BW2 is bounded. 1 p qp q “ ¨p´ q p q Thus W2 C A . An analogous proof to that given in [Ma1, Proposition 4.2] and [Ma1, P p q 1 Proposition 3.3] reveals that W2 C A . R up q 3.2. Establishing the Mourre estimate for H wrt. A. The following Lemma applies to the one-dimensional Laplacian (d 1). “ Lemma 3.1. [Ma1, Lemma 3.4] Recall that E˘ k : 2 2 cos k 2 . Let E 0, 4 E˘ k . Then there exists ε ε E 0 such that for allp θq “C8˘R supportedp { q on I P rE sztε, E p εqu, “ p q ą P c p q “ p ´ ` q θ ∆ Wθ˜ ∆ 0. Thus θ ∆ BW θ ∆ is a compact operator. p q p q“ p q p q LAPFORDISCRETESCHRÖDINGEROPERATOR 10

The following Lemma applies to the multi-dimensional Laplacian (d 2). ě Proposition 3.2. [Ma1, Proposition 4.5] Let 4 4 cos k 2 for k 0,π (3.8) E k : ´ p { q P p q and µ H : 0, E k 4d E k , 4d . p q “ 4 4 cos k 2 for k π, 2π p q “ p p qq Y p ´ p q q # ` p { q P p q For each E µ H there exists ε ε E 0 such that for all θ C8 R supported on I : P p q “ p q ą P c p q “ E ε, E ε , θ ∆ Wθ˜ ∆ 0. In particular, θ ∆ BW θ ∆ is a compact operator. p ´ ` q p q p q“ p q p q Putting together everything discussed in this Section we get the Mourre estimate for H: Proposition 3.3. Let d 1, H ∆ V W and µ H be as defined (1.13). Suppose that V satisfies (1.11) for some mě N and“ any`r, q` 0, , notp bothq zero. Then H C1 A , and for any P P r 8q P p q I µ H , there are γ 0 and compact K such that the Mourre estimate EI H H, iA EI H Ť p q ą p qr s˝ p qě γEI H K holds. In particular the conclusion of Proposition 1.1 holds on the interval I. p q` Proposition 3.3 is basically a reformulation of Proposition 1.1. The proof is easy and goes along the lines of [Ma1, Proposition 3.5]. The decay of the eigenfunctions is a consequence of [Ma2, Theorem 1.5], while the absence of eigenvalues is a consequence of [Ma2, Theorem 1.2]. Note that if H ∆ V , with V as in Proposition 3.3 and W 0, then the above shows that we have a Mourre“ estimate` for H on any I 0, 4d Θ ∆ . “ Ť r sz p q 3.3. Establishing the Mourre estimate for H2 wrt. A. The following Lemma applies to all dimensions (d 1). ě Lemma 3.4. For each E 0, 4d 2d there exists ε ε E 0 such that for all θ C8 R P r szt u “ p q ą P c p q supported on I : E ε, E ε , θ ∆ σθ ∆ 0. In particular, θ ∆ BW2 θ ∆ is a compact operator. “ p ´ ` q p q p q “ p q p q Proof. The relation ∆σ σ 4d ∆ holds. This entails θ ∆ σθ ∆ σθ ∆ θ 4d ∆ for all θ C8 R . Also, θ ∆ θ “4d p ∆´ q0 whenever supp θ p 0q, 2dp orq“ 2pd, 4qdp. So´ itq suffices P c p q p q p ´ q “ p q Ť r q Ť p s to take ε sufficiently small. The compactness of θ ∆ BW2 θ ∆ is then a consequence, together p q p q with the fact that Ui, ∆ is compact.  r s Putting together everything discussed above we get the Mourre estimate for H2:

Proposition 3.5. Let d 1, H2 ∆ V W2 and µ H2 : 0, 4d Θ ∆ 2d . Sup- pose that V satisfies (1.11ě) for some“ m` `N and anypr, qq “0 r , ,sz not p p bothq Y zero.t uq Then 1 P P r 8q H2 C A , and for any I µ H2 , there are γ 0 and compact K such that the Mourre P p q Ť p q ą estimate EI H2 H2, iA ˝EI H2 γEI H2 K holds. In particular the conclusion of Propo- sition 1.2 holdsp qr on the intervals p Iq. ě p q` Proposition 3.5 is basically a reformulation of Proposition 1.2. All the comments done after Proposition 3.3 apply here to H2 as well. 4. Operator norm bounds for functions of self-adjoint operators 4.1. Weighted resolvent bounds with logarithmic terms. Let Q be a self-adjoint operator in a Hilbert space H, z C. The estimate Q z ´1 Im z ´1 is widely known. The aim of this sub-section is to obtainP bounds of the}p form´ fq Q} ďQ | zp ´q|1 f Re z Im z ´1 for some specific classes of functions f : R` R` containing} p logarithmicqp ´ q terms.}ď p Inp whatqq| followsp q| we state results for a general self-adjoint operatorÞÑ Q. The results of this sub-section are later applied in 2 ways : 1) in the next 2 sub-sections (which are still in a general framework), and 2) in the proof of Theorem 1.3 (our Schrödinger problem). In the latter application, Lemma 4.3 is applied with Q A, the discrete version of the generator of dilations, s 1 2, and pk both positive and negative“ for 0 k m 1. “ { ď ď ` LAPFORDISCRETESCHRÖDINGEROPERATOR 11

The next Lemma has already been proved, e.g. [DG, G, GJ2]. We propose a proof by contradiction which lends itself reasonably well to other types of functions. Lemmas 4.2 and 4.3 extend Lemma 4.1 by including logarithmic functions. Finally, Proposition 4.4 is a very nice generalization of Lemmas 4.2 and 4.3 to a wider class of sub-multiplicative functions. The proof is entirely due to Thierry Jecko and we are indebted to him for allowing us to publish it. Lemma 4.1. Let Q be a self-adjoint operator. Let Ω : x,y R2 : 0 y c x , for some c 0. Then for every 0 s 1 there exists C 0 such“ thattp forq P all z ăx | |ďiy Ωx yu: ą ď ď ą “ ` P (4.1) Q s Q z ´1 C x s y ´1. }x y p ´ q }ď x y | | Proof. If s 0 one can take C 1. Suppose that s 0. By the spectral theorem, “ “ ą 1 Q s Q z ´1 2 sup t 2s t x 2 y2 ´ . }x y p ´ q } ď tPR x y p ´ q ` ` ˘ Let t 2s y2 fs x,y,t : x y , defined on x,y,t Ω R Λ, p q “ x 2s t x 2 y2 p q P ˆ z x y p ´ q ` where Ω x,y R2 : y c x and Λ : x,y,t Ω R : y 0,x t . To prove the “ tp q P | | ď x yu “ tp q P ˆ “ “ u Lemma, it is enough to show that fs is uniformly bounded on the entire region Ω R. Suppose ˆ that there is a sequence xn,yn,tn Ω R such that fs xn,yn,tn . Since we have p q P ˆ p qÑ`8 y2 (4.2) 0 1, x,y,t R R˚ R, ď t x 2 y2 ď @p q P ˆ ˆ p ´ q ` and since s 0, we infer that tn xn , as n . In particular, since s 1, we have są x y{x yÑ8 Ñ 8 ĺ tn xn tn xn for n large enough and tn , as n . Next, we have: px y{x yq ďx y{x y | |Ñ8 Ñ8 y2 2 n t xx y2 c (4.3) 0 f x ,y ,t n n . s n n n x y 2 2 2 2 2 ď p qď xn xtny tn xn yn ď tn xn x y 2 2 xxny xtny ´ xtny ` xxny xtny ´ xtny ´ ¯ ´ ¯ We infer:

lim sup fs xn,yn,tn c, nÑ8 p qď which is a contradiction.  Lemma 4.2. Let Q and Ω be as in Lemma 4.1. For every 0 s 1 and p R there exists C 0 such that for all z x iy Ω : ă ă P ą “ ` P (4.4) lnp 1 Q Q s Q z ´1 C lnp 1 x x s y ´1. p `x yqx y p ´ q ď p `x yqx y | | Proof. The case p› 0 is covered by the previous› lemma. By the spectral theorem, ›“ › 2 1 lnp 1 Q Q s Q z ´1 sup ln2p 1 t t 2s t x 2 y2 ´ . p `x yqx y p ´ q ď tPR p `x yqx y p ´ q ` › › ` ˘ Therefore,› we consider the function › ln2p 1 t t 2s y2 gs x,y,t : p `x yq x y , defined on x,y,t Ω R Λ, p q “ ln2p 1 x x 2s t x 2 y2 p q P ˆ z p `x yq x y p ´ q ` where Ω x,y R2 : y c x and Λ : x,y,t Ω R : y 0,x t . We aim “ tp q P | | ď x yu “ tp q P ˆ “ “ u at proving that gs is uniformly bounded on the entire region Ω R. Suppose that there is a ˆ LAPFORDISCRETESCHRÖDINGEROPERATOR 12 sequence xn,yn,tn Ω R such that gs xn,yn,tn . Recalling (4.2), since s 0, up to a subsequence,p we inferq P thatˆ p qÑ `8 ą 2p ln 1 tn either 1 2p p `x yq , q ln 1 xn Ñ8 p `x yq tn or 2 x y , q xn Ñ8 x y 1 as n . We start with case 2). Given α 0, there are Cα,C 0 such that a, b R we have Ñ8 ą α ą @ P 1`xay ln 1 a ln 1`xby 1 1 a (4.5) 0 p `x yq 1 ln `x y 1 ď ln 1 b “ ln ´1 b ¯ ` ď ln 2 1 b ` p `x yq p `x yq α p q ˆ α`x y ˙ 1 a 1 a Cα `x y C x y . ď 1 b ď α b ˆ `x y ˙ ˆ x y ˙ Recalling s 0, 1 and by considering the cases p 0 and p 0 separately, by choosing α small enough, thereP p is aq finite constant C such that ą ă

0 gs xn,yn,tn Cf1 xn,yn,tn , ď p qď p q where fs is the function in Lemma 4.1. Repeating (4.3), we obtain a contradiction. We turn to the case 1). If p 0, (4.5) ensures that 1`xtny , as n . Hence the case ą 1`xxny Ñ 8 Ñ 8 2) holds which in turn gives a contradiction. Suppose now that 1) holds and p 0. Unlike before, we obtain this time that xxny , as ă xtny Ñ 8 n . Using (4.2) and (4.5), there is C,s1 0 such that Ñ8 ą s1 tn 0 gs xn,yn,tn C x y 0, ď p qď xn Ñ ˆx y˙ as n . This is a contradiction.  Ñ8 Lemma 4.3. Let Q and Ω be as in Lemma 4.1. For every 0 s 1 and p0, ..., pm,pm`1 R there exists C 0 such that for all z x iy Ω : ă ă t uĂ ą “m ` P m (4.6) Q s Q z ´1 lnpm`1 Q lnpk Q C x s y ´1 lnpm`1 x lnpk x . x y p ´ q m`1 px yq k px yq ď x y | | m`1 px yq k px yq › k“0 › k“0 › ź › ź Proof. Thanks› to (4.5), the proof is analogous to› that of Lemma 4.2.  We now display Thierry Jecko’s more general result. Let I 1, . We assume 3 things: ` “ r `8q A1: Let ψd : I R 0 be a continuous and decreasing (non-increasing) function such ‚ ÞÑ zt u ´ε that for all ε 0, the function I x ψd x 1 ψd x x is bounded. ą ` Q ÞÑ p p q` { p qq A2: Let ψu : I R 0 be a continuous and increasing (non-decreasing) function such ‚ ÞÑ zt u ´ε that for all ε 0, the function I x ψu x 1 ψu x x is bounded. ą Q ÞÑ p p q` { p qq 2 A3: ψu is sub-multiplicative, that is, there is M 0 such that for all x,y I , ‚ ą p q P ψu xy Mψu x ψu y . p qĺ p q p q ` 2s Now let s 0, 1 and define f : I R 0 by f x : x ψu x ψd x . P p q ÞÑ zt u p q “ p q p q Proposition 4.4. Let Q and Ω be as in Lemma 4.1. In addition to the assumptions A1, A2 and A3 concerning ψd and ψu, suppose there is R0 1 such that f is increasing on R0, . Then ľ r `8q there is C 0 such that f Q Q z ´1 C f x y ´1 uniformly in z x iy Ω. ą } px yqp ´ q }ĺ px yq| | “ ` P m`1 2pk m`1 2pk Remark 4.1. Letting ψu ax ln x aand ψd x ln x immediately p q“ k“0,pką0 k p q p q“ k“0,pkă0 k p q yields Lemma 4.3. On a side note, to satisfy the assumption A3 it is probably also necessary to ś ś have a larger constant in the Napierian logarithm, i.e. instead of ln1 1 x : ln 1 x and p ` q “ p ` q LAPFORDISCRETESCHRÖDINGEROPERATOR 13 lnk x : ln 1 lnk´1 x one should take ln1 µ x : ln µ x and lnk x : ln µ lnk´1 x withp µq “0.p Such` adjustmentp qq is inconsequentialp ` in thisq “ article.p ` q p q “ p ` p qq " Proof. First we choose an analytic extension of f such that c 1 4 in the definition of Ω in Lemma 9.1. ĺ { In this proof C 0 denotes a generic constant that does not depend on x,y and t. Let 2 ą 2s ΩR : x,y R : 0 y c x R . Let fu x : x ψu x . Then fu is sub-multiplicative, “ tp q P ă | |ď´2 x yĺ u p q “ p q increasing on I and x x fu x is bounded by A2. For y 0, let ÞÑ p q ‰ f t f t g t : px yq , and h t : px yq . p q “ t x 2 y2 p q “ t x 2y´2 1 p ´ q ` p ´ q ` Of course, g t h t y´2. To prove the proposition, it is enough by functional calculus to prove p q“ p q ´2 that there is C 0 such that for all x,y Ω, sup R g t Cf x y , or equivalently, ą p q P tP p qĺ px yq (4.7) sup h t Cf x . tPR p qĺ px yq 2s 2s ´2ε First we note that assumptions A1 and A2 imply that f x x ψu x ψd x Cεx x for any ε 0 and so lim f x as x . In turn,p thisq “ impliesp q (togetherp q ľ with the fact ą p q “ `8 Ñ `8 1 that f is continuous) that there is R1 R0 such that max1 x1 R0 f x f x for all x R1. ľ ĺ ĺ p q ĺ p q ľ In Part 1, the estimate (4.7) is proved for x,y ΩR for any finite R (with the constant C p q P depending on R). In Part 2, the estimate (4.7) is proved for x,y Ω Ω2R1 . p q P z PART 1: Let R 1 and x,y ΩR be arbitrarily given. For t 4R, t x t 2 and t x 2y´2 1 ą t2 4y2 p 1q Pt2 4y2 . So | | ľ | ´ | ľ | |{ p ´ q ` ľ {p q` ľ {p q f t f t t 2 f t t 2 h t px yq4y2 4R2 px yq x y 4R2 px yq x y C, p qĺ t2 ĺ t 2 t2 ĺ t 2 t2 ĺ › ›8› ›8 x y › x y › › › where C grows with R. By continuity of the function h›one gets› h› t › C uniformly in t R. › › ›p q ĺ› 1 P Finally, 1 f being uniformly bounded by assumption, suptPR h t C f x f x . { p qĺ px yq 8 ¨ px yq PART 2: Let x,y Ω with x 2R1. › › p q P | |ľ 2 2 › › 2 2 2 Case 1: For t R satisfying t x x 2, one has t 1 t 1› 2 t ›x 2x 4 x , ‚ P | ´ | ĺ | |{ x y “ ` ĺ ` p ´ q ` ĺ x y and so t 2 x . Also, t x x t x 2 R1. Since 2 x t R1 and f is increasing x yĺ x y | | ľ | |´| ´ | ľ | |{ ľ x yľx yľ on R0, , it follows that r 8q h t f t f 2 x fu 2 x ψd 2 x Cfu x ψd x Cf x . p qĺ px yq ĺ p x yq “ p x yq p x yq ĺ px yq px yq “ px yq We used that ψd is decreasing and ψu is sub-multiplicative in the last inequality. Case 2: For t R satisfying t x x 2, one has ‚ P | ´ | ľ | |{ t 2 1 2 t x 2 2x2 1 2 t x 2 8 t x 2 16 t x 2. x y ĺ ` p ´ q ` ĺ ` p ´ q ` p ´ q ĺ x ´ y One further subdivides into 2 subcases. ‚ Subcase 2.1: If t x , then x max 2R1, t . So, | | ĺ | | x yľ p x yq h t f t max f t1 max f t1 2f x , p qĺ px yq ĺ xt1yĺR0 px yq ` R0ĺxt1yĺxxy px yq ĺ px yq where we used the assumption about R1 and that f is increasing on R0, . r 8q ‚ Subcase 2.2: If t x , then 2R1 x x t 4 t x . So f t f 4 t x . | | ľ | | ĺ | |ĺx2 ´2 yĺx y ĺ x ´ y px yq ĺ p x ´ yq This leads to h t f 4 t x 1 t x y . Next, since c 1 4 and R1 1, one has p qĺ p x ´ yq{p ` p ´ q q ĺ { ľ t x y y 4c x 4 x 2 t x and then x ´ y I. | | ĺ | |{p qĺx y{ ĺ | |{ ĺ | ´ | y P x y Since ψd is decreasing, x 4 t x , ψd is decreasing, and fu is sub-multiplicative, we have: x yĺ x ´ y fu 4 t x ψd 4 t x fu t x ψd x t x fu y ψd x h t p x ´ yq p x ´ yq C px ´ yq px yq Cfu x ´ y px yq px yq . p qĺ 1 t x 2y´2 ĺ 1 t x 2y´2 ĺ y 1 t x 2y´2 ` p ´ q ` p ´ q ˆ x y ˙ ` p ´ q LAPFORDISCRETESCHRÖDINGEROPERATOR 14

Finally, since we have t x t x y y , 1 t x 2y´2 t x y 2, and y x , we conclude: x ´ yĺxp ´ q{ y¨x y ` p ´ q “ xp ´ q{ y x yĺx y fu t x y fu t x y h t C pxp ´ q{ yqfu x ψd x C pxp ´ q{ yq fu x ψd x Cf x . p qĺ t x y 2 px yq px yq ĺ t x y 2 px yq px yq ĺ px yq xp ´ q{ y › xp ´ q{ y ›8 › ›  › › › › 4.2. A Heinz inequality for logarithmic functions. Let T,S be arbitrary self-adjoint and bounded from below in a Hilbert space H. The notation T S means that Dom S 1{2 ď r| | s Ă Dom T 1{2 and ψ,Tψ ψ,Sψ for all ψ Dom S 1{2 . The Heinz inequality, cf. [Ka1, H], statesr| | s x yďx y P r| | s (4.8) 0 T S T α Sα, for all 0 α 1, ď ď ñ ď ď ď The aim of this sub-section is to extend this transitive property to a family of (poly)logarithmic functions. In what follows we consider general positive self-adjoint operators T,S and derive results in greater generality. Concerning the detailed application we make of it, notably for the proof of Theorem 1.3, please see sub-section 4.4. In this sub-section we now always suppose 1 T S and hence Dom S1{2 Dom T 1{2 . By the Heinz inequality, T 2α S2α, 0 α ď 1 2ď. But this is equivalentr s to Ă sayingr thats T αψ Sαψ for all ψ Domď Sα . Sinceď Sαďis bijective,{ it follows that T αS´αψ ψ , for} all}ď}ψ H. Thus} T αS´αP is anr elements of B H . In what follows this small} argument}ď} is used} P p q ˚ repeatedly. The Helffer-Sjöstrand formula is also used, see Appendix 9. Let Φn, n N , be the α α P polylogarithmic functions from (8.3). Denote Φ x : Φn x . np q “ p p qq Lemma 4.5. For all 0 α 1 2, Φα T Φ´α S B H , n N˚. ď ď { np q ¨ n p q P p q P ˚ Proof. As discussed in Section 8 the Φn, n N , are Nevanlinna functions that continue P analytically across 1, from C` to C´. Since 1 T S, Theorem 7.3 implies Φn T p´ `8q 2α 2α ď ď p q ď Φn S . By the Heinz inequality, Φ T Φ S , 0 α 1 2. The result follows.  p q n p qď n p q ď ď { α ´α ˚ Lemma 4.6. For all k N and 0 α 1 2, Φ lnk T Φ lnk S B H , n N . P ď ď { np p qq ¨ n p p qq P p q P Proof. Φn lnk x belongs to P 1, because it is the composition of functions all belonging p p qq p´ `8q to P 1, . Also, 1 T S. By Loewner’s Theorem, Φn lnk T Φn lnk S . The result followsp´ from`8q the Heinz inequality.ď ď p p qq ď p p qq  Lemma 4.7. For all 0 p, lnp 1 T ln´p 1 S B H . ď p ` q ¨ p ` q P p q Remark 4.2. For 0 p 1 2, the result follows from well-known results on the logarithm, see e.g. [RS1, Exercise 51ď of Chapterď { VIII]. For p 1 2 we don’t know how prove the result without the use of the polylogarithmic functions. ą { Proof. Write p nα, where n N˚ and 0 α 1 2. Then lnnα 1 T ln´nα 1 S equals “ P ď ď { p ` q p ` q lnnα 1 T Φ´α T Φα T Φ´α S Φα S ln´nα S . p ` q n p q np q n p q np q p q bounded by (8.4) bounded by Lemma 4.5 bounded by (8.4)  The next lemma extends the previous to iterated logarithms. Lemma 4.8. For all k N and 0 p, lnp T ln´p S B H . P ď k p q ¨ k p q P p q Proof. The statement is trivial for k 0 and k 1 is Lemma 4.7. Now let k 1. Write p nα, where n N˚ and 0 α 1 2.“ Then lnnα “T ln´nα S equals ě “ P ď ď { k p q k p q nα ´α α ´α α ´nα ln T Φ lnk 1 T Φ lnk 1 T Φ lnk 1 S Φ lnk 1 S ln S . k p q n p ´ p qq np ´ p qq n p ´ p qq np ´ p qq k p q bounded by (8.4) bounded by Lemma 4.6 bounded by (8.4)  LAPFORDISCRETESCHRÖDINGEROPERATOR 15

The next Lemma extends the previous to products of iterated logarithms. We require extra technical regularity or a much more specific assumption.

Lemma 4.9. Let p0, ..., pm,pm 1 0, and m N be given. Suppose either t ` u Ă r `8q P (a) m`1 ln´pk S C1 T , or ‚ k“0 k p q P p q (b) T Q for some self-adjoint operator Q and m`1 ln´pk S C1 Q . ‚ ś“x y k“0 k p q P p q Then ś m`1 m`1 lnpk T ln´pk S B H . k p q ¨ k p q P p q k“0 k“0 ź ź Proof. By induction on m. The base case m 0 is Lemma 4.7 (recall that ln1 x : ln 1 x ). For the inductive step, m m 1, we have “ p q “ p ` q Ñ ` m`1 m`1 lnpm`2 T lnpk T ln´pm`2 S ln´pk S . m`2 p q k p q ¨ m`2 p q k p q k“0 k“0 ź ź After commuting lnpm`2 T with m`1 ln´pk S this operator is equal to : m`2 p q k“0 k p q m`1 śm`1 lnpk T ln´pk S lnpm`2 T ln´pm`2 S k p q k p q ¨ m`2 p q m`2 p q k“0 k“0 ź ź bounded by Lemma 4.8 bounded by the Induction hypothesis

m`1 m`1 pk pm`2 ´pk ´pm`2 lnk T lnm`2 T , lnk S lnm`2 S . ` p q p q p q ˝ p q k“0 k“0 ź ” ź ı to be developed It is enough to show that the latter term with the under bracket is a bounded operator. If we are assuming a then the term is developed as follows : p q ˜ m`1 m`1 i f pk ´1 ´pk ´1 B lnk T z T T, lnk S z T dz dz, 2π C z p q p ´ q p q ˝p ´ q ^ k“0 k“0 ż B ź ” ź ı pm`2 Sε R where f x lnm`2 x , ε 0, see (9.1). By the assumption a the commutator in the middlep q of “ the integralp q P BpHq .@ Toą keep things simple, one can use Lemmap q 4.1 to get P p q m`1 lnpk T z T ´1 c T δ z T ´1 c x δ y ´1 k p q p ´ q ď x y p ´ q ď x y | | k“0 › ź › › › › › › › for some δ 0 and›c 0. Thus the latter› integral converges in norm to a bounded operator. If we are assumingą b however,ą then the term is developed as follows : p q ˜ m`1 m`1 i h pk ´1 ´pk ´1 B lnk Q z Q Q, lnk S z Q dz dz, 2π C z px yq p ´ q p q ˝p ´ q ^ k“0 k“0 ż B ź ” ź ı pm`2 Sε R where h x lnm`2 x , ε 0. By the assumption b the commutator in the middle of the integralp q“ B Hpx.yq The P restp q is@ asą before. p q  P p q LAPFORDISCRETESCHRÖDINGEROPERATOR 16

4.3. From a LAP in T to a LAP in S. In this sub-section we continue assuming T,S are arbitrary self-adjoint operators satisfying 1 T S. Suppose we have a LAP of the form (1.15) and a local decay estimate (1.16) where theď weightsď are in T . Here we derive estimates that allow to pass to those same estimates with weights in S instead. We state results in greater generality. Concerning the detailed application we make of it please see sub-section 4.4. The next Lemma extends Lemma 4.7.

Lemma 4.10. Fix σ 0, 1 2 and suppose either P r { s (a) S´σ C1 T , or ‚ (b) T PQ pforq some self-adjoint operator Q and S´σ C1 Q . ‚ “x y P p q Then for all 0 p, T σ lnp 1 T S´σ ln´p 1 S B H . ď p ` q ¨ p ` q P p q Proof. The case σ 0 is covered by Lemma 4.7 so we assume σ 0, 1 2 . Since T σS´σ B H p ´p“ P pσ {p s ´σ P p q and ln 1 T ln 1 S B H , it is enough to prove that T ln 1 T ,S ˝ B H . As in thep proof` q of Lemmap ` q4.9 P wep proceedq slightly differently dependingr p on` whetherq sa Por pb qis assumed. Let us treat the case a only. The formal commutator T σ lnp 1 T ,S´σ p isq equalp q to p q r p ` q s ˜ i f σ ´1 ´σ ´1 B T z T T,S ˝ z T dz dz, 2π C z p ´ q r s p ´ q ^ ż B p ε ´σ where f x ln 1 x S R , ε 0. By assumption T,S ˝ B H . Applying Lemma 4.1 showsp thatq “ thisp integral` q P convergesp q @ ą in norm to a boundedr operastor.P p q  The next Lemma extends Lemma 4.9.

Lemma 4.11. Let σ 0, 1 2 , p0, ..., pm,pm 1 0, and m N be given. Suppose either P r { s t ` u Ă r `8q P (a) S´σ m`1 ln´pk S C1 T , or ‚ k“0 k p q P p q (b) T Q for some self-adjoint operator Q and S´σ m`1 ln´pk S C1 Q . ‚ “xś y k“0 k p q P p q Then ś m`1 m`1 T σ lnpk T S´σ ln´pk S B H . k p q ¨ k p q P p q k“0 k“0 ź ź Proof. By induction on m. The base case m 0 is Lemma 4.10. The inductive step is handled “ pm`2 ´σ m`1 ´pk in the same way as in Lemma 4.9. One commutes lnm`2 T with S k“0 lnk S . One applies the inductive hypothesis to the first term, and forp thqe second term (the onep withq the commutator) it is enough to show that ś

m`1 m`1 σ pk pm`2 ´σ ´pk B H T lnk T lnm`2 T ,S lnk S . p q p q p q ˝ P p q k“0 k“0 ź ” ź ı To this end one performs a similar integral expansion as in the proof of Lemma 4.9, with the pm`2 pm`2 same functions, f x lnm`2 x or h x lnm`2 x depending on whether a or b is assumed. Then, top keepq “ thingsp simple,q onep q can “ use thepx factyq that there are δ 0 andp qc p0 q(by Lemma 4.1) such that ą ą

m`1 T σ lnpk T z T ´1 c T σ`δ z T ´1 c x σ`δ y ´1. k p q p ´ q ď x y p ´ q ď x y | | k“0 › ź › › › › › › › One sees that the› integral converges in norm› to a bounded operator.  LAPFORDISCRETESCHRÖDINGEROPERATOR 17

4.4. The 2 previous sub-sections in application. We are now in a position to explain how the results of the preceding 2 sub-sections are applied in the proof of Theorem 1.3. Let A and N be respectively the discrete generator of dilations and the position operator. A and N are unbounded self-adjoint operators. The unboundedness is seen directly from the xgraphy normx y and using an appropriate sequence of unit vectors. The goal is to apply the results of sub-sections 4.2 and 4.3, chiefly Lemmas 4.9 and 4.11, to T,S A , N . However, this cannot be done as such. Instead what we do is to first p q “ px y x yq establish the inequality 1 A ?cd N , see Lemma 4.13 below. We then apply Lemmas 4.9 ďx yď x y and 4.11 to T,S A , ?cd N . Only then do we see that their conclusions remain valid for T,S Ap , Nq“px. y x yq p Forq“px our Schrödingery x yq problem, in application we shall need the conclusion of Lemma 4.9 to hold for T,S A , N , pk 1 for 0 k m and pm`1 1, see for example Lemma 5.3. To pass fromp q a “px LAP withy x weightsyq ” in A toď weightsď in N , weą shall also need the conclusion of x y x y Lemma 4.11 to hold for T,S A , N , σ 1 2, pk 1 2 for 0 k m and pm`1 1 2. In the 2 previous sub-sectionsp q“px we havey x yq kept“ the{ options” open{ for 2ď assumptionsď : a ąor {b , cf. Lemmas 4.9 and 4.11. In application we choose b , i.e. we set T Q where Qp q A,p theq generator of dilations. Thus we must verify the regularityp q criteria : “ x y “

Lemma 4.12. Let σ 0, 1 2 , p0, ..., pm,pm`1 0, and m N be arbitrary. Then m 1 ´p P r { s t u Ă r `8q P N ´σ ` ln k N C1 A . x y k“0 k px yq P p q Proof. śThis is a simple application of Proposition 2.1, (3.3) and the mean value theorem.  Finally, we must give the key inequality that allows to apply Lemmas 4.9 and 4.11 to T,S p q“ A , ?cd N : px y x yq 3 2 2 α α 2 α Lemma 4.13. Let cd : max d d 1, 4 d 1 . 0 α 1, A 1 cd N 1 . “ p ` ` p ` qq @ ď ď p ` q ď p q p ` q Proof. The result is not new but we repeat the proof for convenience. Let ψ Dom N 2 Dom A2 . First we calculate for α 1 : P r s Ă r s “ d 2 2 2 2 2 ψ, A 1 ψ ψ Aψ ψ ψ 2 Njψ p ` q “} } `} } ď} } ` } }` } } „ j“1  @ D ÿ d d 2 2 2 2 2 2 ψ d ψ 4 Njψ d 1 cd ψ Njψ ď} } ` } } ` j“1 } } p ` qď } } ` j“1 } } „ ÿ  „ ÿ  2 cd ψ, N 1 ψ . “ p ` q Pass to exponent α by invoking@ the HeinzD inequality (4.8).  5. Preliminary lemmas for the Proof of Theorem 1.3 Let P denote the orthogonal projection onto the pure point spectral subspace of H. A denotes the discrete generator of dilations (1.12). Please note that throughout all this Section we state the results for the Hamiltonian H, but they also apply to H2 with the same proof (just exchange W with W2 and W˜ with σ). Proposition 5.1. [GJ1] u, v Dom A , the rank one operator u v : ψ v, ψ u C1 A . @ P r s | yx | Ñx y P p q Lemma 5.2. Assume V satisfies (1.10) and (1.11) for some m N, and r, q 0, not both 8 R P P r `8q zero. Then for any open interval I µ H and any η Cc with supp η I, P EI H and P Kη H C1 A . Ă p q P p q p q Ă p q p q P p q Proof. I µ H so the Mourre estimate holds for H and A on I. In particular the point Ă p q spectrum of H is finite on I, see [ABG, Corollary 7.2.11]. Therefore P EI H is a finite rank p q LAPFORDISCRETESCHRÖDINGEROPERATOR 18 operator. By [Ma2, Theorem 1.5], the eigenfunctions of H, if any, belong to the domain of A. 1 K We may therefore apply Proposition 5.1 to get P EI H C A . As for P η H it is equal to 1 p q P p q p q η H P EI H η H , and so belongs to C A .  p q´ p q p q p q N 8 R Lemma 5.3. Assume V satisfies (1.10) for some q r 2, m . Let η Cc be supported on an open interval I. Then 0 p q 4, M mą, “ P P p q @ ď ă { ě (5.1) η H η ∆ w2p,1 A , p p q´ p qq M p q (5.2) w2p,1 A η H η ∆ w2p,1 A M p qp p q´ p qq M p q are compact operators. Proof. We prove the first one and leave the second one for the reader. By Proposition 9.3, ∆ C1 w2p,1 A , since w2p,1 x Sε R , ε 0, and ∆ C1 A . Thus ∆, w2p,1 A B H . P p M p qq M p q P p q @ ą P p q r M p qs˝ P p q By the Helffer-Sjöstrand formula and the resolvent identity, η H η ∆ w2p,1 A equals p p q´ p qq M p q i η˜ ´1 ´1 2p,1 B z H V W z ∆ wM A dz dz 2π C z p ´ q p ` qp ´ q p q ^ ż B i η˜ ´1 2p,1 ´1 B z H V W wM A z ∆ dz dz “ 2π C z p ´ q p ` q p qp ´ q ^ ż B i η˜ ´1 ´1 2p,1 B z H V W z ∆ , wM A ˝dz dz ` 2π C z p ´ q p ` qrp ´ q p qs ^ ż B

i η˜ ´1 2p,1 2p,1 ´2p,´1 2p,1 ´1 B z H V wM N W wM N wM N wM A z ∆ dz dz “ 2π C z p ´ q ¨ p q ` p q˛ p q p qp ´ q ^ ż B compact by (1.10) compact bounded by Lemma 4.9 ˚ ‹ i η˜ ´˝1 ´1 2p,1 ‚ ´1 B z H V W z ∆ ∆, wM A ˝ z ∆ dz dz. ` 2π C z p ´ q p ` qp ´ q r p qs p ´ q ^ ż B compact The integrands of the last two integrals are compact operators. With the support of η bounded, the integrals converge in norm, and so the compactness is preserved in the limit.  Lemma 5.4. Let R 1 and recall that wα,β A R is given by (1.14). Consider ą M p { q K˜0 η ∆ V, iA η ∆ η H η ∆ H, iA η ∆ η H H, iA η H η ∆ , “ p qr s˝ p q ` p p q´ p qqr s˝ p q` p qr s˝p p q´ p qq and K0 : K˜0 η ∆ W, iA η ∆ K˜0 η ∆ KW η ∆ η ∆ BW η ∆ , “ ` p qr s˝ p q“ ` p q p q` p q p q 8 R Θ where η Cc is supported on an open interval I 0, 4d ∆ . Assume V satisfies as- sumptionsP (1.10p )qand (1.11) for some q r 2, m ŤN. r Thensz forp allq 1 2 p q 4, M m, 2p,1 ˜ 2p,1 ą “ P { ă ă { ě wM A R K0wM A R is a compact operator whose norm is uniformly bounded w.r.t. R. Furthermore,p { q if I pµ H{ ,q then further shrinking the size of the interval I also allows 2p,1 2p,Ă1 p q w A R K0w A R to be a compact operator whose norm is uniformly bounded w.r.t. R. M p { q M p { q 2p,1 2p,1 Proof. Write w A R K˜0w A R as M p { q M p { q 2p,1 A ´2p,´1 ´2p,´1 2p,1 A 2p,1 2p,1 w w A K˜ w A w , K˜ : w A K˜0w A . M R M p q M p q M R “ M p q M p q ˆ ˙ ˆ ˙ bounded uniformly in R bounded uniformly in R Thus we want to show that K˜ is a compact operator. K˜ is the sum of the following 3 terms : 2p,1 2p,1 K˜1 : w A η ∆ V, iA η ∆ w A , “ M p q p qr s˝ p q M p q LAPFORDISCRETESCHRÖDINGEROPERATOR 19

˜ 2p,1 2p,1 K2 : wM A η H η ∆ H, iA ˝η ∆ wM A , “ 2p,1p qp p q´ p qqr s p q 2p,1p q K˜3 : w A η H H, iA η H η ∆ w A . “ M p q p qr s˝p p q´ p qq M p q Each of these terms are compact, as explained below. 2p,1 1 2p,1 For K˜1 : we want to commute w A with η ∆ . Since η ∆ C A and w x ‚ M p q p q p q P p q M p q P Sε R , ε 0, w2p,1 A , η ∆ B H by Proposition 9.3. Also, w2p,1 A w´2p,´1 N B H p q @ ą r M p q p qs˝ P p q M p q M p q P p q by Lemma 4.9, and w2p,1 N V, iA w2p,1 N is a compact operator, by the assumption (1.11) M p qr s˝ M p q (1 2 p q 4). Thus K˜1 is a compact operator. { ă ă { For K˜2 : Write it as ‚ w2p,1 A η H η ∆ ∆, iA η ∆ w2p,1 A w2p,1 A η H η ∆ V, iA η ∆ w2p,1 A . M p qp p q´ p qqr s˝ p q M p q` M p qp p q´ p qqr s˝ p q M p q 2p,1 For the first term commute ∆, iA ˝η ∆ with wM A and then apply (5.2). For the second 2rp,1 s p q p q term commute η ∆ with wM A and then apply (5.1) and assumption (1.10). We leave the details to the reader.p q p q For K˜3 : same idea as for K˜2. ‚ Now the second part of the statement where we assume I µ H . Recall KW and BW are given by (3.4) and (3.5). It is enough to show that Ă p q 2p,1 2p,1 K˜4 : w A η ∆ KW η ∆ w A “ M p q p q p q M p q and 2p,1 2p,1 K˜5 : w A η ∆ BW η ∆ w A “ M p q p q p q M p q ˜ 2p,1 2p,1 2p,1 are compact. For K4, commute wM A with η ∆ and use the fact that wM A KW wM A is p q 2p,p1 q ˚ 2pp,1q p q compact. For K˜5, it is enough to prove that w A η ∆ UiW˜ S Si η ∆ w A is compact. M p q p q p i ´ q p q M p q By Lemma 3.1 and Proposition 3.2, η ∆ W˜ η ∆ 0 if the the support of η is sufficiently small. 2p,1 p q p q“˜ ˚ 2p,1 So it is enough to prove that wM A η ∆ ,Ui ˝W Si Si η ∆ wM A is compact. This can be gleaned from the following groupingp qr p q s p ´ q p q p q 2p,1 ´2p,´1 2p,1 2p,1 ´2p,´1 ˚ 2p,1 w A w N w N η ∆ ,Ui w N w N W˜ S Si η ∆ w A . M p q M p q M p qr p q s˝ M p q M p q p i ´ q p q M p q bounded by Lemma 4.9 compact bounded by Lemma 4.9 

6. Proof of Theorem 1.3 We are now ready to prove the projected weighted Mourre estimate (2.3), which in turn will imply the LAP (2.4). The proof makes use of almost analytic extensions of C8 R bounded functions and the class of functions Sρ R with ρ 0, see Appendix 9. We also mentionp q that the proof is essentially the same as that ofp [GJ2q , Theorem“ 4.15] (see also [Ma1, Theorem 5.4]), but we display it in detail for the reader’s convenience. Please note that throughout all this Section we work with the Hamiltonian H, but the same developments work for H2.

1

0

Figure 1. From left to right : EJ x , θ x , η x , χ x , EI x . p q p q p q p q p q Proof of Theorem 1.3. E µ H so let I µ H be an open interval containing E such that the P p q Ă p q K 1 Mourre estimate holds over I. By Lemma 5.2, P EI H and P η H are of class C A , for any η C8 R with supp η I. Let θ,η,χ C8 R bep bumpq functionsp q such that ηθ p θq, χη η P c p q p q Ă P c p q “ “ LAPFORDISCRETESCHRÖDINGEROPERATOR 20 and supp χ I, see Figure 1. Initially J, θ and η are chosen and fixed so that one may apply Lemma 3.1p qĂand Proposition 3.2. As for χ and I they are chosen to fulfill the conditions drawn in Figure 1. Later in the proof we will shrink the interval I around E (and thereby possibly J as well). We aim to derive (2.3) on I for B ϕ A R , C ϕ1 A R , some R 1 and ϕ given by “ p { q “ p { q ą t R R dx a R (6.1) ϕ : , ϕ : t 2p,1 , t , p 1 2. ÞÑ ÞÑ ´8 x w x P ą { ż x y M p q Note that ϕ S0 R , so that ϕ A R B H for all R 1. Please note that it is sufficient to P p q p { q P p q ą obtain the LAP (1.15) for all 1 2 p p0, for some p0 1 2, and only M m, because then the extension to all p 1 2 and{ Mă mĺ is automatic. Considerą { the bounded“ operator ą { ľ F : P Kθ H H, iϕ A R θ H P K “ p qr p { qs˝ p q i 1 ϕ˜ K ´1 ´1 K B z P θ H z A R H, iA ˝ z A R θ H P dz dz. “ 2π R C z p q p qp ´ { q r s p ´ { q p q ^ ż B To save room, henceforth we drop the element dz dz. By Lemma 5.2 P Kη H C1 A , so ^ p q P p q P Kη H , z A R ´1 z A R ´1 P Kη H , A R z A R ´1. r p q p ´ { q s˝ “ p ´ { q r p q { s˝p ´ { q Next to P Kθ H we introduce P Kη H and commute it with z A R ´1. We get : p q p q p ´ { q i 1 ϕ˜ K ´1 K K ´1 K F B z P θ H z A R P η H H, iA ˝η H P z A R θ H P “ 2π R C z p q p qp ´ { q p qr s p q p ´ { q p q ż B 1 1 1 1 A ´ 2 ´p,´ A B ´p,´ A A ´ 2 P Kθ H w 2 1 w 2 θ H P K, ` p q R M R ¨ R2 ¨ M R R p q A E ˆ ˙ ˆ ˙ A E where B1 B H is uniformly bounded in R. We refer to [Ma1, Proof of Theorem 5.4] for extra details. NextP p toq each η H we insert χ H , and then decompose η H H, iA η H as follows : p q p q p qr s˝ p q η H H, iA η H η ∆ ∆, iA η ∆ K0, p qr s˝ p q“ p qr s˝ p q` where

K0 η ∆ V W, iA η ∆ η H η ∆ H, iA η ∆ η H H, iA η H η ∆ . “ p qr ` s˝ p q ` p p q´ p qqr s˝ p q` p qr s˝p p q´ p qq Note that K0 is the compact operator that appears in Lemma 5.4. We also let K K M0 : P χ H η ∆ ∆, iA η ∆ χ H P . “ p q p qr s˝ p q p q Thus :

i 1 ϕ˜ K ´1 ´1 K F B z P θ H z A R M0 z A R θ H P “ 2π R C z p q p qp ´ { q p ´ { q p q ż B i 1 ϕ˜ K ´1 K K ´1 K (6.2) B z P θ H z A R P χ H K0χ H P z A R θ H P ` 2π R C z p q p qp ´ { q p q p q p ´ { q p q ż B 1 1 1 1 A ´ 2 ´p,´ A B ´p,´ A A ´ 2 P Kθ H w 2 1 w 2 θ H P K. ` p q R M R ¨ R2 ¨ M R R p q A E ˆ ˙ ˆ ˙ A E We write the second term in (6.2) as R´1 A R ´1{2w´p,´1{2 A R Kw´p,´1{2 A R A R ´1{2 x { y M p { q M p { qx { y where 1 1 1 1 i ϕ˜ A 2 p, A p, A A 2 K 2 ´1 K K ´1 2 : B z wM z A R P χ H K0χ H P z A R wM . “ 2π C z p q R R p ´ { q p q p q p ´ { q R R ż B A E ˆ ˙ ˆ ˙ A E CLAIM. LAPFORDISCRETESCHRÖDINGEROPERATOR 21

K is a compact operator for 1 2 p q 4, where q is the exponent that appears in (1.10) and (1.11) – Hypothesis (H). Also{ theă normă { of K goes to zero as the support of χ gets tighter around E, and this uniformly in R. PROOF OF CLAIM. Write K as

1 1 i ϕ˜ A 2 p, A K ´ ´ 2 ´1 B z wM z A R “ 2π C z p q R R p ´ { q ż B A E ˆ ˙ bounded by Lemma 4.3 ą 1 1 2p,1 A K K ´1 p, 2 A A 2 w P χ H K0χ H P z A R w . M R p q p q p ´ { q M R R ˆ ˙ ˆ ˙ A E bounded by Lemma 4.3 2p,1 A K K C1 2p,1 Sε R Now we want to commute wM R with P χ H . Since P χ H A and wM x , ε 0, p q p q P p q p q P p q @ ą ` ˘ 2p,1 A 2p,1 A K B H wM wM , P χ H R R p q ˝ P p q ˆ ˙ ” ˆ ˙ ı by Proposition 9.3. So K K1 K2, where “ `

1 1 i ϕ˜ A 2 p, A K ´ ´ 2 ´1 1 B z wM z A R “ 2π C z p q R R p ´ { q ż B A E ˆ ˙ bounded by Lemma 4.3 ą 1 1 K 2p,1 A K ´1 p, 2 A A 2 P χ H w K0χ H P z A R w p q M R p q p ´ { q M R R ˆ ˙ ˆ ˙ A E bounded by Lemma 4.3 and 1 3 i ϕ˜ A 2 3p, A K ´ ´ 2 ´1 2 B z wM z A R “2π C z p q R R p ´ { q ż B A E ˆ ˙ bounded by Lemma 4.3 ą 1 1 2p,1 A 2p,1 A K K ´1 p, 2 A A 2 wM wM , P χ H K0χ H P z A R wM . R R p q ˝ p q p ´ { q R R ˆ ˙ ” ˆ ˙ ı ˆ ˙ A E bounded uniformly in R by Proposition 9.3 bounded by Lemma 4.3 By Lemma 4.3, there is C 0 such that ą 1 3 3 A 2 ´3p,´ A ´3p,´ 1 w 2 z A R ´1 Cw 2 x x 2 y ´1, R M R p ´ { q ď M p qx y | | ›A E ˆ ˙ › and › › › 1 1 › 1 A 2 p, A p, 1 w 2 z A R ´1 Cw 2 x x 2 y ´1. R M R p ´ { q ď M p qx y | | ›A E ˆ ˙ › Now apply Lemma 4.3› and (9.2) with ρ 0 and ℓ › 2. Note that K2 is bounded above by a › K “ “› } } multiple of K0χ H P times } p q } 3 1 ´3p,´ 1 p, 1 dxdy ρ´1´ℓ ℓ 2 2 ´1 2 2 ´1 x y wM x x y wM x x y dxdy 2p,1 , Ω x y | | ¨ p qx y | | ¨ p qx y | | “ Ω x 2w x ă8 ż ż x y M p q thanks to p 1 2. Because K0 is compact, it follows that K2 is a compact operator and K2 ą { } } goes to zero as the support of χ gets tighter around E. We repeat a similar operation for K1 : LAPFORDISCRETESCHRÖDINGEROPERATOR 22

1 1 p, ´p,´ write w 2 A w2p,1 A w 2 A and commute w2p,1 A with χ H P K. Doing this gives M R “ M R M R M R p q ` ˘ ` ˘ ` ˘ ` ˘ 1 1 i ϕ˜ A 2 p, A K ´ ´ 2 ´1 1 B z wM z A R “2π C z p q R R p ´ { q ż B A E ˆ ˙ bounded by Lemma 4.3 ą 1 1 K 2p,1 A 2p,1 A K ´1 ´p,´ 2 A A 2 P χ H w K0w χ H P z A R w K3 . p q M R M R p q p ´ { q M R R ` ˆ ˙ ˆ ˙ ˆ ˙ A E compact compact by Lemma 5.4 bounded by Lemma 4.3

K K 2p,1 2p,1 One shows that 1 c P χ H wM A R K0wM A R for some constant c 0 and goes } }ď } p q p { q p { q} 2p,1 ą2p,1 to zero as the support of χ gets tighter around E, notably because wM A R K0wM A R is K K pK { q 2p,1 p { q compact by Lemma 5.4. As for 3, it is compact and satisfies 3 c P χ H wM A R K0 for some constant c 0. All in all, we see that K is a compact} operator}ď } suchp q that Kp {goesq to} zero as the support ofą χ gets tighter around E, uniformly in R. This proves the claim.} }

We now proceed with the proof of the Theorem. Thanks to the claim we have :

i 1 ϕ˜ K ´1 ´1 K F B z P θ H z A R M0 z A R θ H P “ 2π R C z p q p qp ´ { q p ´ { q p q ż B 1 1 1 1 A ´ 2 ´p,´ A B K ´p,´ A A ´ 2 P Kθ H w 2 1 w 2 θ H P K. ` p q R M R R2 ` R M R R p q A E ˆ ˙ˆ ˙ ˆ ˙ A E

´1 The next thing to do is to commute z A R with M0 : p ´ { q

i 1 ϕ˜ K ´2 K F B z P θ H z A R M0θ H P “ 2π R C z p q p qp ´ { q p q ż B i 1 ϕ˜ K ´1 ´1 K B z P θ H z A R M0, z A R ˝θ H P ` 2π R C z p q p qp ´ { q r p ´ { q s p q ż B 1 1 1 1 A ´ 2 ´p,´ A B K ´p,´ A A ´ 2 P Kθ H w 2 1 w 2 θ H P K. ` p q R M R R2 ` R M R R p q A E ˆ ˙ˆ ˙ ˆ ˙ A E We apply (9.3) to the first integral (which converges in norm), while for the second integral we 1 use the fact that M0 C A to conclude that there exists a B2 B H whose norm is uniformly bounded in R such thatP p q P p q

´1 K 1 K F R P θ H ϕ A R M0θ H P “ p q p { q p q 1 1 1 1 A ´ 2 ´p,´ A B K ´p,´ A A ´ 2 P Kθ H w 2 2 w 2 θ H P K. ` p q R M R R2 ` R M R R p q A E ˆ ˙ˆ ˙ ˆ ˙ A E

Now ϕ1 A R A R ´1w´2p,´1 A R . By Proposition 9.3, p { q“x { y M p { q

1 1 1 1 ´ ´p,´ 2 p, 2 ´1 1 A R 2 w A R , M0 A R 2 w A R R B rx { y M p { q s˝x { y M p { q“ LAPFORDISCRETESCHRÖDINGEROPERATOR 23

1 for some B B H whose norm is uniformly bounded in R. Thus there is B3 B H with norm uniform in RP suchp q that P p q 1 1 ´ 1 1 ´ ´1 K A 2 ´p,´ 2 A ´p,´ 2 A A 2 K F R P θ H w M0w θ H P “ p q R M R M R R p q ˆ ˙ ˆ ˙ A 1 E 1 1 A E 1 A ´ 2 ´p,´ A B K ´p,´ A A ´ 2 P Kθ H w 2 3 w 2 θ H P K ` p q R M R R2 ` R M R R p q ˆ ˙ˆ ˙ ˆ ˙ A E 1 1 A E 1 1 A ´ 2 ´p,´ A ´p,´ A A ´ 2 γR´1P Kθ H w 2 P Kχ H η2 ∆ χ H P Kw 2 θ H P K ě p q R M R p q p q p q M R R p q ˆ ˙ ˆ ˙ A1 E 1 1 1 A E A ´ 2 ´p,´ A B K ´p,´ A A ´ 2 P Kθ H w 2 3 w 2 θ H P K ` p q R M R R2 ` R M R R p q A E ˆ ˙ˆ ˙ ˆ ˙ A E where γ 0 comes from applying the Mourre estimate for ∆ and A. Let ą K˜ : γP Kχ H η2 ∆ η2 H χ H P K. “ p qp p q´ p qq p q Note that K˜ is compact with K˜ vanishing as the support of χ gets tighter around E. Thus } } 1 1 1 1 A ´ 2 ´p,´ A ´p,´ A A ´ 2 F γR´1P Kθ H w 2 P Kχ H η2 H χ H P Kw 2 θ H P K ě p q R M R p q p q p q M R R p q ˆ ˙ ˆ ˙ A1 E 1 A E ´ 1 K K˜ 1 ´ K A 2 ´p,´ 2 A B3 ´p,´ 2 A A 2 K P θ H wM 2 ` wM θ H P . ` p q R R ˜R ` R ¸ R R p q A E ˆ ˙ ˆ ˙ A E 1 ´p,´ 1 K 2 K K 2 K 2 ´ 2 Finally, we commute P χ H η H χ H P P η H P with wM A R A R , and see that p q p q p q “ p q p { qx { y 1 1 ´p,´ 1 p, 1 K 2 K 2 ´ 2 2 2 ´1 2 γ P η H P , wM A R A R ˝wM A R A R R B 2 r p q p { qx { y s p { qx { y “ for some B B H whose norm is uniformly bounded in R. Thus there is B4 B H with norm uniformP in Rp suchq that P p q A ´1 A F γR´1P Kθ H w´2p,´1 θ H P K ě p q R M R p q ˆ ˙ A1 E 1 ´ 1 K K˜ 1 ´ K A 2 ´p,´ 2 A B4 ´p,´ 2 A A 2 K P θ H wM 2 ` wM θ H P . ` p q R R ˜R ` R ¸ R R p q A E ˆ ˙ ˆ ˙ A E To conclude, we shrink the support of χ to ensure that K K˜ γ 3 and choose R 1 so that } ` }ă { ą B4 R γ 3. Then K K˜ γ 3 and B4 R γ 3, so } }{ ă { ` ě´ { { ě´ { ´1 K A K γ K A ´2p,´1 A K (6.3) F P θ H H, iϕ θ H P P θ H wM θ H P . “ p q R ˝ p q ě 3R p q R R p q ˆ ˙ ˆ ˙ 1 ” 1 ı A E Let J be any open interval with J I. Applying EJ1 H on both sides of this inequality yields 1 Ť p q 1 ´p,´ ´ 2 2 the projected weighted Mourre estimate (2.3), with c γ 3R and C A R wM A R , 1 2 p q 4. The proof of Theorem 1.3 is complete,“ as{p explainedq in“x Section{ y2.3. p { q { ă ă { 7. Appendix A. Nevanlinna functions, operator monotone functions and Loewner’s theorem We revisit Loewner’s theorem on matrix operator monotone functions, see e.g. [L], [Do], [Si2] and [Ha]. This wonderful theorem makes a striking connection between the operator monotone functions and the Nevanlinna functions. Let C` (resp. C´) denote the complex numbers with strictly positive (resp. negative) imaginary part. A Nevanlinna function (also known as Herglotz, Pick or R function) is an analytic function LAPFORDISCRETESCHRÖDINGEROPERATOR 24 on C` that maps C` to C`. A function f : C` C is Nevanlinna if and only if it admits a representation ÞÑ

1 λ C` (7.1) f z α βz 2 dµ λ , z p q“ ` ` R λ z ´ λ 1 p q P ż ˆ ´ ` ˙ 2 ´1 where α R, β 0, and µ is a positive Borel measure on R satisfying R λ 1 dµ λ . We referP to [Doě, Theorem 1 of Chapter II] for a proof of this wonderfulp result` q. Thep integralqă8 representation is unique. The measure µ is recovered from f by the Stieltjesş inversion formula 1 λ2`δ µ λ1, λ2 lim lim Im f λ iε dλ. pp sq “ δÓ0 εÓ0 π p p ` qq żλ1`δ Standard examples of Nevanlinna functions given in the literature include zp for 0 p 1, zp for 1 p 0, and the logarithm ln z with the branch cut , 0 . ď ď ´ ´ ď ď p q p´8 s Notation. Let a, b be an open interval (finite or infinite). Denote P a, b the set of Nevan- p q p q linna functions that continue analytically across a, b into C´ and where the continuation is by reflection. p q

Functions in P a, b are real-valued on a, b and their measure satisfies µ a, b 0, see [Do, Lemma 2, Chapterp q II]. Functions in P pa, bq are strictly increasing on a, b pp, unlessqq “ they are constant. Indeed, if f is not a constant functionp q β and µ cannot be simulaneouslyp q zero and so

1 dµ λ f x β p q 2 0, x a, b . p q“ ` R λ x ą P p q ż p ´ q Let Mn C be the set of n n matrices with entries in C, and consider a function f : a, b R. Definition.p q f is matrix monotoneˆ of order n in a, b if f T f S holds wheneverp T,Sq ÞÑ in p q p q ď p q Mn C are hermitian matrices with spectrum in a, b and T S. p q p q ď In 1934 Karl Loewner proved the following remarkable theorem that characterizes the matrix monotone functions : Theorem 7.1. [L] Let f : a, b R, where a, b is a finite or infinite open interval. Then f is matrix operator monotonep of orderq ÞÑ n in a, bp forq all n N if and only if f admits an analytic continuation that belongs to P a, b . p q P p q Loewner’s theorem is a truly wonderful result and has been reproved in several different ways, see e.g. [Do, Theorem I of Chapter VII and Chapter IX for the converse], [Si2] or [Ha] and refer- ences therein for a concise historical exposition. For the purpose of this article, we need a version of Loewner’s theorem that applies to unbounded self-adjoint operators. In B. Simon’s book [Si2, Chapter 2] it is explicitly discussed how Loewner’s theorem extends to unbounded operators. We propose below yet another proof of the extension to the semi-bounded operators (which is the case for A and N ). For self-adjoint operators T , S which are bounded from below, the in- x y x y equality T S means Dom S 1{2 Dom T 1{2 and ψ,Tψ ψ,Sψ for all ψ Dom S 1{2 . ď r| | sĂ r| | s x yďx y P r| | s Definition. f is operator monotone in a, b R if f T f S holds whenever T,S (possibly unbounded) are self-adjoint operators inpH withqĂ spectrump qď containedp q in a, b and T S. p q ď Assuming f P a, b , let µ be the measure associated to f, see (7.1), and denote the supremum P p q of the support of µ by Σµ. In what follows the discussion is for general self-adjoint T,S, but one should keep in mind that the results are applied to T A and S ?cd N , cf. Lemma 4.13. We start with a Lemma : “x y “ x y LAPFORDISCRETESCHRÖDINGEROPERATOR 25

Lemma 7.2. Let f belong to P a, . Then p `8q (1) Σµ a, (2) givenď T a self-adjoint operator with inf σ T a 0, we have f T bounded from below and Dom T Dom f T with equalityp p ifqq and ą onlyą if β 0. p q r sĎ r| p q|s ą Proof. For (1), we start by noting that f belongs to P a, so f admits an integral repre- p `8q sentation as in (7.1). Thus Σµ a holds thanks to the Stieltjes inversion formula and the fact that f is real-valued on a, ď. For (2), adding a constant to f does not alter the assumptions of the Lemma and leadsp to`8q the same conclusion, so we may assume f inf σ T 0, where inf σ T a (recall f is non-decreasing on a, ). We start by showingp thatp p limqqqf ąx x β, as xp p qq ą, or equivalently p `8q p q{ “ Ñ `8 Σµ 1 λx lim ` dµ λ 0. xÑ`8 x λ x λ2 1 p q“ ż´8 p ´ qp ` q We wish to exchange the order of the limit and integration. We have (7.2) 1 λx x´1 λ x ´1 1, λ, x , 0 1, 0, a a a2 1, . p ` q p ´ q ď @p q P p´8 s ˆ r `8q Yr s ˆ r ` ` `8q We mayˇ apply the dominatedˇ convergence theorem, and the above limit follows.a This limit impliesˇ that f x x is a boundedˇ function on a, and hence Dom T Dom f T . For the reverse inclusion,p q{ x f x is a well defined boundedp `8q function on inf σr Ts Ă, iffr| βp q|s0.  We are now ready to{ provep q the extension of Theorem 7.1 to semi-boundedp p p qq `8q operatorsą : Theorem 7.3. Let f : a, b R, where 0 a b . Then f is operator monotone in a, b if and only if f admitsp q an ÞÑ analytic continuationă ă ď that `8 belongs to P a, b . p q p q Proof. If f is operator monotone in a, b , then in particular it is matrix operator monotone of order n in a, b for all n N, and sopf admitsq an analytic continuation that belongs to P a, b by Loewner’sp theorem.q ThisP direction is in fact the hard direction in Loewner’s theroem,p butq the extension is trivial! For the converse, we suppose that f admits an analytic continuation that belongs to P a, b . Consider the two separate cases b and b . If b , then f is matrix operatorp q monotone of order n in a, b for allăn `8N by Loewner’s“ `8 theorem.ă `8 But, since the interval a, b is finite, this is equivalentp toq being operatorP monotone in a, b , see e.g. [BS, Lemma 2.2] orp [Si2q, p q Chapter 2]. Now the case b . We have Σµ a by Lemma 7.2, and “ `8 ď Σµ 1 λ f x α βx dµ λ , x a, . p q“ ` ` λ x ´ λ2 1 p q P p `8q ż´8 ˆ ´ ` ˙ Let Σµ 1 λ (7.3) gr x : α dµ λ , g x : f x βx, x a, . p q “ ` λ x ´ λ2 1 p q p q “ p q´ P p `8q ż´r ˆ ´ ` ˙ Note that gr rPR` is of a sequence of functions of the real variable x that converges pointwise t u 1 to g x as r for all x a, . Moreover, since for every fixed r, gr x 0, all functions in thep q sequenceÑ `8 are increasingP p in`8q the variable x. We need a Lemma. p qě

Lemma 7.4. For every fixed ε 0, the sub-sequence gr x has the property that gr x ą t p qurą1{ε p qÕ g x for every x max Σµ ε, a . p q ą p ` q Proof. Clearly gr x g x as r for all x a, . What needs to be shown is that p q Ñ p q Ñ `8 P p `8q the sequence gr x is increasing pointwise. Let ε 0 be given and fix x max Σµ ε, a . Recall a is assumed top beq strictly positive. The integrandą in (7.3) is equal to ą p ` q 1 λx ` . λ x λ2 1 p ´ qp ` q LAPFORDISCRETESCHRÖDINGEROPERATOR 26

On the one hand, x Σµ ε implies that λ x is negative for all λ supp µ. On the other hand, the assumptioną x `a 0 implies thatp ´1 q λx is negative for allP λ 1 x. Thus, for ą ą p ` q ă ´ { all x max Σµ ε, a and for all λ 1 a, the integrand in (7.3) is strictly positive. Thus, as r increasesą p above` 1 qa, the value ofă the ´ integral{ in (7.3) strictly increases. This completes the proof of the Lemma. {  Continuation of the proof of Theorem 7.3. Now fix T,S self-adjoint operators with spectrum contained in a, and T S. Let ψ H. T S implies ψ, λ T ´1ψ ψ, λ S ´1ψ p `8q ď P ď x p ´ q yďx p ´ q y for all λ Σµ, e.g. [Ka, Theorem VI.2.21]. We integrate over the compact r, Σµ : ď r´ s Σµ Σµ ´1 ´1 ψ, λ T ψ dµ λ ψ, λ S ψ dµ λ , for every finite r Σµ . x p ´ q y p qď x p ´ q y p q ą | | ż´r ż´r Moreover, since the integrand is norm continuous, we infer

(7.4) ψ, gr T ψ ψ, gr S ψ , for every finite r Σµ . x p q yďx p q y ą | | Next we choose in Lemma 7.4 ε small enough so that max Σµ ε, a inf σ T inf σ S . p ` q ă p p qq ď p p qq We obtain gr x g x (for r 1 ε) as r goes to infinity. In turn, the monotone conver- p q Õ p q ą { gence theorem for forms, e.g. [Ka, Theorem VIII.3.11], ensures that ψ, gr T ψ converges to x p q y ψ, g T ψ as r for all ψ Dom g T 1{2 , and similarly for S. Taking limits in (7.4) x p q y Ñ `8 P r| p q| s gives Dom g S 1{2 Dom g T 1{2 and r| p q| sĂ r| p q| s ψ, g T ψ ψ, g S ψ , ψ Dom g S 1{2 . x p q yďx p q y P r| p q| s Thus, if β 0 we have “ ψ,f T ψ ψ,f S ψ , ψ Dom g S 1{2 Dom f S 1{2 , x p q yďx p q y P r| p q| s“ r| p q| s whereas if β 0, we use the fact that 0 T S, and Dom S1{2 Dom f S 1{2 , Dom T 1{2 ą ă ď r s“ r| p q| s r s“ Dom f T 1{2 , by Lemma 7.2, which yields r| p q| s ψ,f T ψ ψ,f S ψ , ψ Dom S1{2 Dom f S 1{2 . x p q yďx p q y P r s“ r| p q| s This gives the result.  To close this Section, we have a remark about results in the literature on order relations for general self-adjoint operators. In [O], Olson introduced the spectral order for self-adjoint operators : T ĺ S if and only if E T E S for all t R. Here E T and p´8,tsp q ě p´8,tsp q P p´8,tsp q E S are the spectral resolutions of the identity for T and S. He showed that T n Sn for p´8,tsp q ď every n N is equivalent to T ĺ S, see also [U, Proposition 5]. Furthermore, it is shown in [FK] that thisP order relation is equivalent to f T f S for any continuous monotone nondecreasing function f defined on an interval whichp containsqď p σq T σ S . For the purpose of this article, n n p qY p q we do not know if A ?cd N holds n N but if it does it would considerably simplify this article. To checkx y thisď inequality p x yq by brute@ forceP seems to be unbearable.

8. Appendix B. Polylogarithms of positive order are Nevanlinna functions That logarithm with the standard branch cut is a Nevanlinna function follows from the identity ln z ln r iθ, where z reiθ, r 0, and θ π,π . The integral representation of the logarithmp q “ p is q` “ ą P p´ q 0 1 λ ln z dλ. p q“ λ z ´ λ2 1 ż´8 ˆ ´ ` ˙ The composition of Nevanlinna functions produces another Nevanlinna function. So for example lnp z is Nevanlinna for 0 p 1. What about higher powers of the logarithm ? Certainly the squarep q and cube of the logarithmď ď are not Nevannlina functions. Indeed writing ln2 z ln2 r θ2 i2θ ln r , and ln3 z ln3 r 3θ2 ln r iθ 3 ln2 r θ2 p q“ p q´ ` p q p q“ p q´ p q` p p q´ q LAPFORDISCRETESCHRÖDINGEROPERATOR 27 reveals that these functions do not map C` to C`. In spite of this there are functions that are Nevanlinna and are "almost" equal to the logarithms. To motivate the idea, we note that the Stieltjes inversion formula gives

λ2`δ 1 2 0 λ1 0, lim lim Im ln λ iε dλ λ2 ě δÓ0 εÓ0 π p ` q “ 2 ln λ dλ λ 0 żλ1`δ # λ1 2 ` ˘ p| |q ď when applied to ln2 z and ş p q

λ2`δ 1 3 0 λ1 0, lim lim Im ln λ iε dλ λ2 2 2 ě δÓ0 εÓ0 π λ1 δ p ` q “ 3 ln λ π dλ λ2 0 ż ` # λ1 p| |q ´ ď 3 ` ˘ when applied to ln z . This suggests to calculateş the` Nevanlinna functions˘ corresponding to p q 2 the measures dµ λ 1 ln λ dλ and dµ λ 1 ln λ dλ. p q“ tλă´1u p´ q p q“ tλă´1u p´ q We introduce polylogarithms. We refer to [Le] and [PBM] for formulas and a detailed exposi- tion. The polylogarithm of order σ C is defined by the power series P 8 zk Liσ z : . p q “ kσ k“1 ÿ The definition is valid for complex z 1 and is extended to the complex plane by analytic continuation. For the purpose of this| article,| ă we are interested in the polylogarithms with σ 2, or σ 3 if we want to simplify by taking the smallest integer above 2. The standard branchą cut “ is 1, for Li1 z and 1, for Li2 z and Li3 z . The polylogarithm of order 1 can be r `8q p q p `8q p q p q written in terms of a logarithm as Li1 z ln 1 z . The polylogarithm of order 2 is called the dilogarithm or Spence function whilep q“´ the polylogarithmp ´ q of order 3 is called the trilogarithm. On p. 494 of [PBM] the following integral representation is given without proof : z 8 lnσ λ (8.1) Liσ 1 z p q dλ, ` p q“ Γ σ 1 λ λ z p ` q ż1 p ´ q for arg 1 z π, Re σ 1, or for z 1, Re σ 0. Here Γ is the Gamma function. Obviously| p ´ (8.1q|) is ă equivalentp qą´ to “ p q ą 8 σ 8 1 ln λ 1 1 λ σ (8.2) Liσ 1 z p q dλ ln λ dλ. ` p q“´Γ σ 1 λ λ2 1 ` Γ σ 1 λ z ´ λ2 1 p q p ` q ż1 p ` q p ` q ż1 ˆ ´ ` ˙ This means that Liσ`1 z is a Nevanlinna function for Re σ 1. Although not difficult to prove, it is not clear wherep q a proof of (8.1) can be found inp theqą´ literature. Thus we prove it : Proposition 8.1. (8.1) is true. Proof. Let λ 1. Writing 1 λ λ z as a power series in z we have 1 λ λ z 8 λ´k´2zk forěz 1. Then the{p rhsp ´ of (qq8.1) is equal to {p p ´ qq “ k“0 | |ă 8 8 8 ř z 8 lnσ λ zk 8 lnσ λ zk p qzkdλ p qdλ , Re σ 1. Γ σ 1 λk`2 “ Γ σ 1 λk`1 “ kσ`1 p qą´ 1 k“0 k“1 1 k“1 p ` q ż ÿ ÿ p ` q ż ÿ To evaluate the last integral the change of variable k ln λ t was performed, followed by the p q “ definition of the Gamma function. Thus the rhs of (8.1) is equal to Liσ`1 z for z 1 and Re σ 1. The result follows by the uniqueness of the analytic continuation.p q | | ă  p qą´ While we’re at it we note that Li0 z z 1 z is also a Nevanlinna function. Definition. For σ C, p q“ {p ´ q P (8.3) Φσ z : Liσ z , z C , 1 . p q “´ p´ q P zp´8 ´ s LAPFORDISCRETESCHRÖDINGEROPERATOR 28

Clearly (8.2) implies that the Φσ are Nevanlinna for Re σ 1 with integral representations given by : p qą´ 8 σ ´1 1 ln λ 1 1 λ σ Φσ 1 z p q dλ ln λ dλ, ` p q“ Γ σ 1 λ λ2 1 ` Γ σ 1 λ z ´ λ2 1 p´ q p ` q ż1 p ` q p ` q ż´8 ˆ ´ ` ˙ for arg 1 z π. Finally, the other reason we resort to polylogarithms is because they decay at the| samep ` rateq| ă as the logarithms, at least for positive integer order (this follows directly from the inversion/reflection formula [Le, (6) of Appendix A.2.7] together with Lin 0 0), namely : p q“ Φn x 1 (8.4) lim p q , n N. xÑ`8 lnn x “ n! P p q 9. Appendix C. Almost analytic extensions and Helffer-Sjöstrand calculus We refer to [D], [DG], [GJ1], [GJ2], [MS] for more details. Let ρ R and denote by Sρ R the class of functions ϕ in C8 R such that P p q p q pkq ρ´k (9.1) ϕ x Ck x , for all k N. | p q| ď x y P Lemma 9.1. [D] and [DG] Let ϕ Sρ R , ρ R. Then for every N Z` and every c 0, there P p q P P ą exists a smooth function ϕ˜N : C C, called an almost analytic extension of ϕ, satisfying: Ñ ϕ˜N x i0 ϕ x , x R; p ` q“ p q @ P

supp ϕ˜N Ω : x iy : y c x ; p qĂ “ t ` | |ď x yu ϕ˜N x iy 0, y R whenever ϕ x 0; p ` q“ @ P p q“

ϕ˜N ρ´1´ℓ ℓ (9.2) ℓ N 0,N , B x iy cℓ x y for some constants cℓ 0. @ P X r s ˇ z p ` qˇ ď x y | | ą ˇ B ˇ ˇ ˇ Now let Q be a self-adjointˇ operator. ˇ ˇ ˇ Lemma 9.2. Let ρ 0 and ϕ Sρ R . Then for all k N and N N: ă P p q P P i k! ϕ˜N (9.3) ϕpkq Q p q B z z Q ´1´kdz dz p q“ 2π C z p qp ´ q ^ ż B where the integral exists in the norm topology. For ρ 0, the following limit exists: ě ˜ i k! ϕθR N (9.4) ϕpkq Q f lim p q Bp q z z Q ´1´kfdz dz, for all f Dom Q ρ . p q “ RÑ8 2π C z p qp ´ q ^ P rx y s ż B In particular, if ϕ Sρ R with 0 ρ k and ϕpkq is a bounded function, then ϕpkq Q is a bounded operator andP (9.3p )q holds (withď theă integral converging in norm). p q Proposition 9.3. [GJ1] Let T be a bounded self-adjoint operator satisfying T C1 Q . Then : P p q (9.5) T, z Q ´1 z Q ´1 T,Q z Q ´1, r p ´ q s˝ “ p ´ q r s˝p ´ q and for any ϕ Sρ R with ρ 1, T C1 ϕ Q and P p q ă P p p qq i ϕ˜N ´1 ´1 (9.6) T, ϕ Q ˝ B z Q T,Q ˝ z Q dz dz. r p qs “ 2π C z p ´ q r s p ´ q ^ ż B LAPFORDISCRETESCHRÖDINGEROPERATOR 29

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Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400, Talence, France Email address: [email protected]

Independent researcher, Jersey City, 07305, NJ, USA Email address: [email protected]