Bound states due to a strong $\delta$ interaction supported by a curved surface

TL;DR

This paper investigates bound states of a Schrödinger operator with a strong delta interaction supported on a curved surface, showing the existence of discrete spectrum and deriving eigenvalue asymptotics based on surface geometry.

Contribution

It demonstrates the existence of bound states for a delta interaction on curved surfaces and provides an asymptotic expansion of eigenvalues considering the surface's geometry.

Findings

Discrete spectrum exists for large enough interaction strength.

Eigenvalues have an asymptotic expansion related to surface geometry.

Bound states are supported by non-planar, asymptotically planar surfaces.

Abstract

We study the Schr\"odinger operator $-\Delta -\alpha \delta (x-\Gamma)$ in $L^2(\R^3)$ with a $\delta$ interaction supported by an infinite non-planar surface $\Gamma$ which is smooth, admits a global normal parameterization with a uniformly elliptic metric. We show that if $\Gamma $ is asymptotically planar in a suitable sense and $\alpha>0$ is sufficiently large this operator has a non-empty discrete spectrum and derive an asymptotic expansion of the eigenvalues in terms of a ``two-dimensional'' comparison operator determined by the geometry of the surface $\Gamma$. [A revised version, to appear in J. Phys. A]