Bound States in Curved Quantum Layers

TL;DR

This paper investigates how the curvature of a surface influences the existence of bound states for a quantum particle confined in a curved layer with hard-wall boundaries, under conditions where surface curvature diminishes at infinity.

Contribution

It provides sufficient conditions for the existence of bound states induced by geometry in quantum layers over non-compact surfaces with vanishing curvature at infinity.

Findings

Bound states exist under specific geometric conditions.

Surface curvature decay at infinity is crucial for bound state formation.

The analysis applies to layers built over surfaces with geodesic polar coordinates.

Abstract

We consider a nonrelativistic quantum particle constrained to a curved layer of constant width built over a non-compact surface embedded in $R^3$. We suppose that the latter is endowed with the geodesic polar coordinates and that the layer has the hard-wall boundary. Under the assumption that the surface curvatures vanish at infinity we find sufficient conditions which guarantee the existence of geometrically induced bound states.