Geometrically induced spectrum in curved leaky wires

TL;DR

This paper investigates how the geometry of curved leaky quantum wires influences their spectral properties, demonstrating that non-straight curves induce at least one bound state below the essential spectrum threshold.

Contribution

It establishes that curved infinite wires support bound states due to geometric effects, extending understanding of spectral behavior in quantum waveguides.

Findings

Curved wires have at least one bound state below the essential spectrum.

Straight wires do not support such bound states.

Geometric curvature induces spectral bound states.

Abstract

We study measure perturbations of the Laplacian in $L^2(\R^2)$ supported by an infinite curve $\Gamma$ in the plane which is asymptotically straight in a suitable sense. We show that if $\Gamma$ is not a straight line, such a ``leaky quantum wire'' has at least one bound state below the threshold of the essential spectrum.