On the spectrum of curved quantum waveguides

TL;DR

This paper studies the spectral properties of the Laplace operator in curved quantum waveguides, analyzing how curvature affects the essential and discrete spectra under various boundary conditions.

Contribution

It provides new conditions for the existence of discrete spectra and bounds on spectral gaps, extending understanding of spectral stability in curved waveguides.

Findings

Essential spectrum is stable under curvature vanishing at infinity.

Conditions for the existence of geometrically induced discrete spectrum.

Lower bounds on the spectral gap for locally curved strips.

Abstract

The spectrum of the Laplace operator in a curved strip of constant width built along an infinite plane curve, subject to three different types of boundary conditions (Dirichlet, Neumann and a combination of these ones, respectively), is investigated. We prove that the essential spectrum as a set is stable under any curvature of the reference curve which vanishes at infinity and find various sufficient conditions which guarantee the existence of geometrically induced discrete spectrum. Furthermore, we derive a lower bound on the distance between the essential spectrum and the spectral threshold for locally curved strips. The paper is also intended as an overview of some new and old results on spectral properties of curved quantum waveguides.