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Bifurcation of double eigenvalues for Aharonov-Bohm operators with a moving pole

Laura Abatangelo, Veronica Felli

Abstract

We study double eigenvalues of Aharonov-Bohm operators with Dirichlet boundary conditions in planar domains containing the origin. We focus on the behavior of double eigenvalues when the potential's circulation is a fixed half-integer number and the operator's pole is moving on straight lines in a neighborhood of the origin. We prove that bifurcation occurs if the pole is moving along straight lines in a certain number of cones with positive measure. More precise information is given for symmetric domains; in particular, in the special case of the disk, any eigenvalue is double if the pole is located at the centre, but there exists a whole neighborhood where it bifurcates into two distinct branches.

Bifurcation of double eigenvalues for Aharonov-Bohm operators with a moving pole

Abstract

We study double eigenvalues of Aharonov-Bohm operators with Dirichlet boundary conditions in planar domains containing the origin. We focus on the behavior of double eigenvalues when the potential's circulation is a fixed half-integer number and the operator's pole is moving on straight lines in a neighborhood of the origin. We prove that bifurcation occurs if the pole is moving along straight lines in a certain number of cones with positive measure. More precise information is given for symmetric domains; in particular, in the special case of the disk, any eigenvalue is double if the pole is located at the centre, but there exists a whole neighborhood where it bifurcates into two distinct branches.

Paper Structure

This paper contains 15 sections, 23 theorems, 157 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An equivalent eigenvalue problem by gauge transformation
  4. Some key quantities
  5. Perturbation theory around the origin along a fixed direction
  6. Application to our setting
  7. Analysis of the restricted quadratic form.
  8. Study of the function Calphau
  9. Decomposition of the eigenspace
  10. Ramification of eigenbranches from double eigenvalues
  11. Applications to symmetric domains
  12. Invariance under a rotation
  13. Axially symmetric domains
  14. Symmetries of the rectangle
  15. The case of the disk

Key Result

Theorem 1.1

(BonnaillieNorisNysTerracini2014, Lena2015) Let $\Omega \subset \mathbb{R}^2$ be open, bounded and connected. Fix any $j \in \mathbb{N} \setminus\{0\}$. The map $a \in \Omega \mapsto \lambda_j^{a}$ has a continuous extension on $\overline{\Omega}$, that is where $\lambda_j$ is the $j$-th eigenvalue of the Laplacian in $\Omega$ with Dirichlet boundary conditions. Moreover, if $b \in \Omega$ and th

Figures (2)

  • Figure 1: Ramification of double eigenvalues.
  • Figure 2: Domains with the symmetries of a rectangle.

Theorems & Definitions (41)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Proposition 2.1
  • Lemma 2.2
  • Theorem 2.3
  • ...and 31 more