Magnetic Dirichlet Laplacian in curved waveguides

TL;DR

The paper studies the magnetic Dirichlet Laplacian on a curved planar waveguide and shows that, when the magnetic field vanishes at infinity, the essential spectrum remains stable and, for small magnetic strength, the discrete spectrum below $\frac{\pi^2}{d^2}$ is absent. This is achieved by transforming the curved strip to a straight domain via a unitary map, deriving a lower bound on the quadratic form through a perturbative decomposition, and employing a Hardy-type inequality to control curvature-induced perturbations. A Weyl sequence constructed on the straight strip demonstrates the inclusion of $[\frac{\pi^2}{d^2},\infty)$ in the essential spectrum under decay assumptions on the magnetic potential. The results generalize previous local perturbation findings to non-local deformations and highlight the magnetic field's ability to suppress geometry-induced bound states, with potential implications for quantum waveguide design. All mathematical notation is contained within $...$ delimiters to ensure precise formalism and applicability in analytical and computational contexts.

Abstract

For a two-dimensional curved waveguide, it is well known that the spectrum of the Dirichlet Laplacian is unstable. Any perturbation of the straight strip produces eigenvalues below the essential spectrum. In this paper, a magnetic field is added. We explicitly prove that the spectrum of the magnetic Laplacian is stable under small but non-local deformations of the waveguide.

Theorems & Definitions (7)

• Theorem 3.1
• proof
• Lemma 3.2
• Theorem 3.3
• proof
• Theorem 4.1
• Theorem 4.2