Pavel Exner, Semjon Vugalter
On the existence of bound states in asymmetric leaky wires}
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ABSTRACT. 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 $\delta$-interaction supported by a non-straight, piecewise smooth curve $\LL$ dividing the plane into two regions of which one, the `interior', is convex. The other interaction component is a constant positive potential $V_0$ in one of the regions. We show that in the critical case, $V_0=lpha^2$, 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, $V_0<lpha^2$, the system may have any finite number of bound states provided the angle between the asymptotes of $\LL$ is small enough.