On the spectrum of leaky surfaces with a potential bias

Abstract

We discuss operators of the type $H=–\Delta +V(x)–\alpha \delta (x–\Sigma )$ with an attractive interaction, $\alpha >0$ in $L^2({\mathbb{R}}^3)$, where $\Sigma$ is an infinite surface, asymptotically planar and smooth outside a compact, dividing the space into two regions, of which one is supposed to be convex, and $VB$ is a potential bias being a positive constant $V_0$ in one of the regions and zero in the other. We find the essential spectrum and ask about the existence of the discrete one with a particular attention to the critical case, $V_0={\alpha }^2$. We show that ${\sigma }_{\mathrm{disc}}(H)$ is then empty if the bias is supported in the ‘exterior’ region, while in the opposite case isolated eigenvalues may exist.