Perturbation of discriminant for one-dimensional discrete Schr\"odinger operator with sparse periodic potential

TL;DR

This paper investigates the spectrum of a one-dimensional discrete Schrödinger operator with sparse periodic potential by representing the discriminant with Chebyshev polynomials and analyzing how perturbations affect the spectrum.

Contribution

It introduces a novel approach of using Chebyshev polynomial representations of the discriminant to study spectral properties under perturbations for complex-valued sparse periodic potentials.

Findings

Spectrum characterized as intersections of algebraic curves

Discriminant perturbation analysis reveals spectral structure

Chebyshev polynomial representation simplifies spectral analysis

Abstract

We consider the one-dimensional discrete Schr\"odinger operator with complex-valued sparse periodic potential. The spectrum for a complex-valued periodic potential is a complicated compact set in the complex plane represented by real intersections of algebraic curves determined by a discriminant. We represent the discriminant by Chebyshev polynomials and use perturbations of the discriminant to study the spectrum.