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Magnetic Dirichlet Laplacian on a perturbed twisted tube

Diana Barseghyan, Ricardo Abreu Blaya, Juan Bory-Reyes, Baruch Schneider

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.

Magnetic Dirichlet Laplacian on a perturbed twisted tube

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.
Paper Structure (6 sections, 5 theorems, 61 equations)

This paper contains 6 sections, 5 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Magnetic Dirichlet Laplacian
  3. Proofs
  4. Proof of Theorem \ref{['main']}
  5. Proof of Theorem \ref{['main1']}
  6. Proof of Lemma \ref{['lemma']}

Key Result

Theorem 2.1

The spectrum of $H_{\Omega_\beta}(0)$ is purely absolutely continuous and covers the half-line $[E,\infty)$.

Theorems & Definitions (5)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Lemma 3.1