On the spectrum of leaky surfaces with a potential bias

  • Pavel Exner

    Czech Academy of Sciences, Řež near Prague, Czechia, and Czech Technical University, Prague, Czechia
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Abstract

We discuss operators of the type H=Δ+V(x)αδ(xΣ)H = –\Delta + V(x) – \alpha \delta (x–\Sigma) with an attractive interaction, α>0\alpha > 0 in L2(R3)L^2(\mathbb R^3), where Σ\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 VBVB is a potential bias being a positive constant V0V_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, V0=α2V_0 = \alpha^2. We show that σdisc(H)\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.