On spectrum of a Schroedinger operator with a fast oscillating compactly supported potential

TL;DR

This paper investigates how fast oscillating, compactly supported potentials affect the emergence of eigenvalues from the essential spectrum in Schrödinger operators, providing conditions for their existence and asymptotic behavior.

Contribution

It introduces new criteria for the existence or absence of eigenvalues caused by oscillating potentials and derives their asymptotic expansion.

Findings

Conditions for eigenvalue emergence are established.

Asymptotic expansion of the eigenvalue is derived.

Criteria for absence of such eigenvalues are provided.

Abstract

We study the phenomenon of an eigenvalue emerging from essential spectrum of a Schroedinger operator perturbed by a fast oscillating compactly supported potential. We prove the sufficient conditions for the existence and absence of such eigenvalue. If exists, we obtain the leading term of its asymptotics expansion.