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

Abstract

We study the Schrödinger operator $-Δ-αδ(x-Γ)$ in $L^2(\R^3)$ with a $δ$ interaction supported by an infinite non-planar surface $Γ$ which is smooth, admits a global normal parameterization with a uniformly elliptic metric. We show that if $Γ$ is asymptotically planar in a suitable sense and $α>0$ 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 $Γ$. [A revised version, to appear in J. Phys. A]

LaTeX 2e, 21 pages