Magnetic Dirichlet Laplacian on deformed waveguides

TL;DR

This paper proves that the spectrum of the magnetic Dirichlet Laplacian in deformed waveguides remains stable under small boundary deformations, contrasting the known instability in non-magnetic cases.

Contribution

It extends spectral stability results to magnetic waveguides without the local boundary deformation restriction.

Findings

Spectrum remains stable under small boundary deformations with magnetic field

Magnetic field alters the spectral instability known in non-magnetic waveguides

No eigenvalues below the essential spectrum are introduced by small deformations

Abstract

It is well known that the spectrum of the Dirichlet Laplacian for a two-dimensional waveguide, which is a local deformation of a straight strip, is unstable with respect to waveguide boundary deformations. This means that, when the waveguide is a straight strip, the spectrum of the Dirichlet Laplacian is purely essential. On the other hand, local boundary perturbations of the straight strip produce eigenvalues below the essential spectrum. This paper considers the Dirichlet-Laplace operator with a compactly supported magnetic field. Furthermore, we omit the condition that the boundary perturbation is local. We prove that, in this case, the spectrum of the magnetic Laplacian is stable under small deformations of the waveguide boundary.