Gap asymptotics in a weakly bent leaky quantum wire

TL;DR

This paper investigates how small bends or deformations in a planar quantum wire with attractive delta interactions affect the bound states, revealing that the eigenvalue gap scales with the fourth power of the bending angle.

Contribution

It provides a precise asymptotic analysis of the eigenvalue gap in weakly bent or deformed leaky quantum wires, a novel insight into geometric effects on quantum bound states.

Findings

Eigenvalue gap scales with the fourth power of bending angle.

Single eigenvalue appears in weakly bent configurations.

Behavior analyzed for both bending and asymptotic wiggles.

Abstract

The main question studied in this paper concerns the weak-coupling behavior of the geometrically induced bound states of singular Schr\"odinger 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.