Gap asymptotics in a weakly bent leaky quantum wire

Abstract

The main question studied in this paper concerns the weak-coupling behavior of the geometrically induced bound states of singular Schrödinger operators with an attractive $\delta$ interaction supported by a planar, asymptotically straight curve $\Gamma$. We demonstrate that if $\Gamma$ is only slightly bent or weakly deformed, then there is a single eigenvalue and the gap between it and the continuum threshold is in the leading order proportional to the fourth power of the bending angle, or the deformation parameter. For comparison, we analyze the behavior of a general geometrical induced eigenvalue in the situation when one of the curve asymptotes is wiggled.

20 pages, no figures

References in corpus (1)

• Leaky Quantum Graphs: A Review

Cited by in corpus (4)

• On the negative spectrum of the Robin Laplacian in corner domains
• Asymptotics of the bound state induced by $\delta$-interaction supported on a weakly deformed plane
• Spectral properties of soft quantum waveguides
• Scattering of particles bounded to an infinite planar curve