Limiting absorption principle for discrete Schrödinger operators with a Wigner-von Neumann potential and a slowly decaying potential Sylvain Golenia, Marc-Adrien Mandich To cite this version: 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.