On the existence of bound states in asymmetric leaky wires

TL;DR

This paper investigates the spectral properties of a quantum leaky wire model with a potential bias, revealing conditions under which bound states exist depending on the potential and geometry of the wire.

Contribution

It provides a detailed analysis of the existence of bound states in asymmetric leaky wires with potential bias, including critical and non-critical cases, and geometric conditions for bound states.

Findings

Bound states exist if the potential bias is supported in the interior in the critical case.

In the subcritical case, the number of bound states can be arbitrarily large with small enough asymptote angles.

The spectral properties depend on the potential bias and the geometry of the wire.

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 $\mathcal{L}$ 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, $V_0=\alpha^2$, 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, $V_0<\alpha^2$, the system may have any finite number of bound states provided the angle between the asymptotes of $\mathcal{L}$ is small enough.