On the spectral properties of non-selfadjoint discrete Schr\"odinger operators

TL;DR

This paper investigates how compact non-selfadjoint perturbations affect the spectral properties of a selfadjoint operator, using advanced mathematical techniques to analyze the discrete spectrum and embedded eigenvalues.

Contribution

It provides a unified framework linking the regularity of perturbations to spectral properties, extending existing results and applying to the one-dimensional discrete Laplacian.

Findings

Analysis of the discrete spectrum structure

Conditions for embedded eigenvalues

Existence of limiting absorption principles

Abstract

Let $H_0$ be a purely absolutely continuous selfadjoint operator acting on some separable infinite-dimensional Hilbert space and $V$ be a compact non-selfadjoint perturbation. We relate the regularity properties of $V$ to various spectral properties of the perturbed operator $H_0+V$. The structure of the discrete spectrum and the embedded eigenvalues are analysed jointly with the existence of limiting absorption principles in a unified framework. Our results are based on a suitable combination of complex scaling techniques, resonance theory and positive commutators methods. Various results scattered throughout the literature are recovered and extended. For illustrative purposes, the case of the one-dimensional discrete Laplacian is emphasized.