On the existence of bound states in asymmetric leaky wires Available to Purchase

We analyze spectral properties of a leaky wire model with a potential bias. It describes a two-dimensional quantum particle exposed to a potential consisting of two parts. One is an attractive δ-interaction supported by a non-straight, piecewise smooth curve $L$ dividing the plane into two regions of which one, the “interior,” is convex. The other interaction component is a constant positive potential V₀ in one of the regions. We show that in the critical case, V₀ = α², the discrete spectrum is non-void if and only if the bias is supported in the interior. We also analyze the non-critical situations, in particular, we show that in the subcritical case, V₀ < α², the system may have any finite number of bound states provided the angle between the asymptotes of $L$ is small enough.