Magnetic Dirichlet Laplacian on a perturbed twisted tube

TL;DR

This paper proves that the spectrum of the magnetic Dirichlet Laplacian on a twisted tube remains stable under small local domain deformations, contrasting with the non-magnetic case where the spectrum is unstable.

Contribution

It demonstrates the spectral stability of the magnetic Dirichlet Laplacian under domain perturbations, providing explicit proof for the magnetic case.

Findings

Magnetic field induces spectral stability under domain deformations

Spectrum remains purely essential for unperturbed twisted tubes

Eigenvalues below the essential spectrum appear only with perturbations in non-magnetic case

Abstract

It is well known that the spectrum of the Dirichlet Laplacian for a compact perturbation of a three-dimensional, periodically twisted tube is unstable with respect to domain deformations. This means that if the periodically twisted tube is unperturbed, then the spectrum of the Dirichlet Laplacian is purely essential. On the other hand, the perturbation of this domain produces eigenvalues below the essential spectrum. This paper considers the Dirichlet-Laplace operator with a magnetic field. We explicitly prove that the spectrum of the magnetic Laplacian is stable under small and local deformations of the domain.