On quantum waveguide with shrinking potential

TL;DR

This paper investigates the spectral properties of a Schrödinger operator in a multi-dimensional cylinder with a shrinking potential, focusing on the emergence of new eigenvalues from the essential spectrum threshold and providing conditions and asymptotic descriptions for these eigenvalues.

Contribution

It offers new criteria for the emergence of eigenvalues from the essential spectrum in Schrödinger operators with shrinking potentials and constructs their asymptotic expansions.

Findings

Conditions for eigenvalue emergence are established.

Asymptotic expansions of eigenvalues are derived.

Eigenvalues can emerge from the spectrum threshold under certain conditions.

Abstract

We consider the spectrum of a Schroedinger operator in a multi-dimensional cylinder perturbed by a shrinking potential. We study the phenomenon of a new eigenvalue emerging from the threshold of the essential spectrum and give the sufficient conditions for such eigenvalues to emerge. If such eigenvalues exist, we construct their asymptotic expansions.