Dislocation Defects and Diophantine Approximation

TL;DR

This paper investigates eigenvalues in spectral gaps of a Schrödinger operator with a periodic potential plus a defect, linking their existence to Diophantine approximation and generalizing Zheludev's theorem.

Contribution

It introduces a homotopy method to count eigenvalues and establishes a connection between eigenvalue distribution and Diophantine approximation, extending previous results.

Findings

Number of eigenvalues in large gaps is typically one.

Existence of multiple eigenvalues depends on Diophantine approximation solutions.

Provides conditions for solvability of the associated Diophantine problem.

Abstract

In this paper we consider a Schrodinger eigenvalue problem with a potential consisting of a periodic part together with a compactly supported defect potential. Such problems arise as models in condensed matter to describe color in crystals as well as in engineering to describe optical photonic structures. We are interested in studying the existence of point eigenvalues in gaps in the essential spectrum, and in particular in counting the number of such eigenvalues. We use a homotopy argument in the width of the potential to count the eigenvalues as they are created. As a consequence of this we prove the following significant generalization of Zheludev's theorem: the number of point eigenvalues in a gap in the essential spectrum is exactly one for sufficiently large gap number unless a certain Diophantine approximation problem has solutions, in which case there exists a subsequence of gaps containing 0,1 or 2 eigenvalues. We state some conditions under which the solvability of the Diophantine approximation problem can be established.