Thresholds and more bands of A.C. spectrum for the Molchanov--Vainberg Schr\"odinger operator with a more general long range condition

TL;DR

This paper investigates the absolutely continuous spectrum of a discrete Schr"odinger operator with long-range potentials on multi-dimensional integer lattices, identifying new spectral bands and thresholds through analytical and numerical methods.

Contribution

It introduces a novel approach using finite linear combinations of conjugate operators to find additional bands of absolutely continuous spectrum for the operator.

Findings

More bands of a.c. spectrum are identified.

An infinite set of spectral thresholds is rigorously characterized.

Numerical evidence supports the existence of new spectral bands.

Abstract

The existence of absolutely continuous (a.c.) spectrum for the discrete Molchanov-Vainberg Schr\"odinger operator $D+V$ on $\ell^2(\mathbb{Z}^d)$, in dimensions $d\geq 2$, is further investigated for potentials $V$ satisfying the long range condition $n_i(V-\tau_i ^{\kappa}V)(n) = O(\ln^{-q}(|n|))$ for some $q>2$, $\kappa \in \mathbb{N}$ even, 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. In this article \textit{finite} linear combinations of conjugate operators are constructed. These lead to more bands of a.c.\ spectrum being found. However, the new bands of a.c. spectrum are justified mainly by graphical evidence because the coefficients of the linear combinations are obtained by numerical polynomial interpolation. At the same time, an infinitely countable set of thresholds is rigorously identified (these will be defined exactly in the article). We conjecture that the spectrum of $D+V$ in dimension 2 is void of singular continuous spectrum, and that consecutive thresholds constitute endpoints of a band of a.c. spectrum.