On the spectrum of leaky surfaces with a potential bias

Abstract

We discuss operators of the type $H = -Δ+ V(x) - αδ(x-Σ)$ with an attractive interaction, $α>0$, in $L^2(\mathbb{R}^3)$, where $Σ$ 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 $V$ 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, $V_0=α^2$. We show that $σ_\mathrm{disc}(H)$ is then empty if the bias is supported in the `exterior' region, while in the opposite case isolated eigenvalues may exist.

12 pages, to appear in Helge Holden Festschrift