Thresholds and more bands of a.c. Spectrum for the discrete Schr{\"o}dinger operator with a more general long range condition

TL;DR

This paper extends the analysis of absolutely continuous spectrum for discrete Schrödinger operators with long-range potentials, constructing multiple conjugate operators to identify more spectral bands and thresholds.

Contribution

It introduces a new method using finite linear combinations of conjugate operators, implemented via polynomial interpolation, to reveal additional absolutely continuous spectrum bands.

Findings

More bands of a.c. spectrum observed due to new conjugate operators

Identification of an infinite set of spectral thresholds

Numerical evidence supports the theoretical framework

Abstract

We continue the investigation of the existence of absolutely continuous (a.c.) spectrum for the discrete Schr\"odinger operator $\Delta+V$ on $\ell^2(\Z^d)$, in dimensions $d\geq 2$, for potentials $V$ satisfying the long range condition $n_i(V-\tau_i ^{\kappa}V)(n) = O(\ln^{-q}(|n|))$ for some $q>2$, $\kappa \in \N$, and all $1 \leq i \leq d$, as $|n| \to \infty$. $\tau_i ^{\kappa} V$ is the potential shifted by $\kappa$ units on the $i^{\text{th}}$ coordinate. The difference between this article and \cite{GM2} is that here finite linear combinations of conjugate operators are constructed leading to more bands of a.c.\ spectrum being observed. The methodology is backed primarily by graphical evidence because the linear combinations are built by numerically implementing a polynomial interpolation. On the other hand an infinitely countable set of thresholds, whose exact definition is given later, is rigorously identified. Our overall conjecture, at least in dimension 2, is that the spectrum of $\Delta+V$ is void of singular continuous spectrum, and consecutive thresholds are endpoints of a band of a.c. spectrum.