Bound States in Soft Quantum Layers

Abstract

We develop a general approach to study three-dimensional Schrödinger 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 Schrödinger 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 Schrödinger operator possesses an infinite number of discrete eigenvalues.