Pavel Exner
On the spectrum of leaky surfaces with a potential bias
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ABSTRACT. We discuss operators of the type $H = -\Delta + V(x) - lpha \delta(x-\Sigma)$ with an attractive interaction, $lpha>0$, 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 $V$ 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, $V_0=lpha^2$. 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.