A complete classification of threshold properties for one-dimensional discrete Schrödinger operators

Abstract

We consider the discrete one-dimensional Schrödinger operatorand V is a self-adjoint operator on ℓ 2 (Z) with a decay property given by V extending to a compact operator from ℓ ∞,−β (Z) to ℓ 1,β (Z) for some β ≥ 1. We give a complete description of the solutions to Hx = 0, and Hx = 4x, x ∈ ℓ ∞,−β (Z). Using this description we give asymptotic expansions of the resolvent of H at the two thresholds 0 and 4. One of the main results is a precise correspondence between the solutions to Hx = 0 and the leading coefficients in the asymptotic expansion of the resolvent around 0. For the resolvent expansion we implement the expansion scheme of Jensen-Nenciu [4,5] in the full generality.

Spectral and scattering properties at thresholds for the Laplacian in a half-space with a periodic boundary condition

“…These results belong to the intersection of two active research topics in spectral and scattering theory. On one hand, resolvent expansions at thresholds (which have a long history, but which have been more systematically developed since the seminal paper of A. Jensen and G. Nenciu [20], see also [11,17,21,28]). On the second hand, representation formulas for the wave operators and their application to the proof of index theorems in scattering theory (see [5,16,23,24,26,27,29] and references therein).…”

Tosio Kato’s work on non-relativistic quantum mechanics: part 2

“…For ν = 1 with exponentially decaying potentials, the behavior was analyzed by Bollé et al [63,64] and, in general, by Jensen-Nenciu [289]. Ito-Jensen [275] discuss Jacobi matrices (discrete ν = 1).…”

Resolvent expansion for the Schrödinger operator on a graph with infinite rays

“…, N, jointed together. Special cases are the discrete full line Z and the discrete half-line N, considered in [IJ1] and [IJ2], respectively. The perturbation V can be a general non-local operator, which is assumed to decay at infinity in an appropriate sense.…”

“…However, a complete analysis taking into account all possible generalized threshold eigenfunctions has been obtained only recently. The first one is in [IJ1] on the discrete full line Z and more recently on the discrete half-line N in [IJ2]. In these papers the authors implement the expansion scheme of [JN1,JN2] in its full generality.…”

“…In these papers the authors implement the expansion scheme of [JN1,JN2] in its full generality. This paper is a generalization of [IJ1,IJ2] to a graph with infinite rays. The strategy is essentially the same as before.…”