On the spectrum of leaky surfaces with a potential bias
Pavel ExnerCzech Academy of Sciences, Řež near Prague, Czechia, and Czech Technical University, Prague, Czechia
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We discuss operators of the type with an attractive interaction, in , where 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 is a potential bias being a positive constant 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, . We show that is then empty if the bias is supported in the ‘exterior’ region, while in the opposite case isolated eigenvalues may exist.