Bound States in Mildly Curved Layers

TL;DR

This paper investigates the existence and asymptotic behavior of bound states for quantum particles in mildly curved layers, deriving the leading order eigenvalue expansion using Birman-Schwinger analysis.

Contribution

It provides the first detailed analysis of weak-coupling asymptotics of bound states in mildly curved layers with rigorous eigenvalue expansion.

Findings

Derived the leading order in the ground-state eigenvalue expansion.

Established conditions on surface curvature decay for bound states.

Applied Birman-Schwinger analysis to quantum layers.

Abstract

It has been shown recently that a nonrelativistic quantum particle constrained to a hard-wall layer of constant width built over a geodesically complete simply connected noncompact curved surface can have bound states provided the surface is not a plane. In this paper we study the weak-coupling asymptotics of these bound states, i.e. the situation when the surface is a mildly curved plane. Under suitable assumptions about regularity and decay of surface curvatures we derive the leading order in the ground-state eigenvalue expansion. The argument is based on Birman-Schwinger analysis of Schroedinger operators in a planar hard-wall layer.