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.