On the spectrum of leaky surfaces with a potential bias

2017/01/23 by Pavel Exner, Exner, Pavel
Mathematics · #Spectral Theory in Mathematical Physics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1701.06288

Abstract

We discuss operators of the type H = -Δ+ V(x) - αδ(x-Σ) with an attractive interaction, α>0, in L²(ℝ³), 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 V₀ 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=α². We show that σ_disc(H) is then empty if the bias is supported in the `exterior' region, while in the opposite case isolated eigenvalues may exist.

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