Bound states in soft quantum layers

TL;DR

This paper introduces a new method for analyzing three-dimensional Schrödinger operators with surface-dependent confining potentials, revealing conditions for the essential spectrum and the existence of infinitely many bound states.

Contribution

It develops a general approach using parallel coordinates to study spectral properties of Schrödinger operators with surface-dependent potentials, including conditions for bound states.

Findings

Estimate on the essential spectrum location for asymptotically planar surfaces

Existence of infinitely many discrete eigenvalues for certain curved surfaces

Application of the method to surfaces of revolution with positive Gauss curvature

Abstract

We develop a general approach to study three-dimensional Schroedinger operators with confining potentials depending on the distance to a surface. The main idea is to apply parallel coordinates based on the surface but outside its cut locus in the Euclidean space. If the surface is asymptotically planar in a suitable sense, we give an estimate on the location of the essential spectrum of the Schroedinger operator. Moreover, if the surface coincides up to a compact subset with a surface of revolution with strictly positive total Gauss curvature, it is shown that the Schroedinger operator possesses an infinite number of discrete eigenvalues.