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

Abstract

We study the Schrödinger operator $-\Delta -\alpha \delta (x-\Gamma )$ in $L^2({\mathbb{R}}^3)$ 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]

LaTeX 2e, 21 pages