On the spectrum of a Schroedinger operator perturbed by a fast oscillating potential

TL;DR

This paper analyzes the spectral properties of a one-dimensional Schrödinger operator with a rapidly oscillating potential, providing explicit formulas for the essential spectrum and detailed asymptotics for eigenvalues and eigenfunctions.

Contribution

It offers a comprehensive analysis of the spectrum, including explicit essential spectrum and asymptotic expansions for eigenvalues and eigenfunctions, for operators with fast oscillating potentials.

Findings

Explicit form of the essential spectrum.

Existence and multiplicity of discrete spectrum.

Asymptotic expansions for eigenvalues and eigenfunctions.

Abstract

We study the spectrum of a one-dimensional Schroedinger operator perturbed by a fast oscillating potential. The oscillation period is a small parameter. The essential spectrum is found in an explicit form. The existence and multiplicity of the discrete spectrum are studied. The complete asymptotics expansions for the eigenvalues and the associated eigenfunctions are constructed.