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Magnetic Neumann problems with Aharonov-Bohm potentials: boundary asymptotics of eigenvalues and splitting phenomena

Veronica Felli, Prasun Roychowdhury, Giovanni Siclari

Abstract

We study a planar magnetic Schrödinger operator with an Aharonov-Bohm vector potential, under Neumann boundary conditions. Through a gauge transformation, the corresponding eigenvalue problem can be formulated in terms of the Laplacian on a fractured domain, where the fracture lies along the segment connecting the pole to its projection on the boundary. As the pole approaches the boundary, we prove that the eigenvalues converge to those of the Neumann Laplacian and the variation exhibits a logarithmic vanishing rate. In the case of multiple eigenvalues, when the pole approaches a fixed point of the boundary, we observe a splitting phenomenon, with the largest branch separating from the others.

Magnetic Neumann problems with Aharonov-Bohm potentials: boundary asymptotics of eigenvalues and splitting phenomena

Abstract

We study a planar magnetic Schrödinger operator with an Aharonov-Bohm vector potential, under Neumann boundary conditions. Through a gauge transformation, the corresponding eigenvalue problem can be formulated in terms of the Laplacian on a fractured domain, where the fracture lies along the segment connecting the pole to its projection on the boundary. As the pole approaches the boundary, we prove that the eigenvalues converge to those of the Neumann Laplacian and the variation exhibits a logarithmic vanishing rate. In the case of multiple eigenvalues, when the pole approaches a fixed point of the boundary, we observe a splitting phenomenon, with the largest branch separating from the others.
Paper Structure (14 sections, 26 theorems, 296 equations, 1 figure)

This paper contains 14 sections, 26 theorems, 296 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. An equivalent formulation via gauge transformation
  4. Mosco type convergence of the complements of segments
  5. Spectral stability
  6. Expansion of eigenvalues
  7. E for an eigenfunction u
  8. The case of simple eigenvalues
  9. Pole approaching a fixed boundary point
  10. An equivalent problem
  11. L2-norm versus energy of potentials
  12. Reduction to the case with constant jump across the segment
  13. The case of ellipses
  14. Proof of Proposition \ref{['p:asyE']}, Proposition \ref{['p:asy-r']}, and Theorem \ref{['t:toward-a-fixed-point']}

Key Result

Proposition 1.1

Under assumptions eq:ass-Omega--hp_p, for every $a\in\Omega$, let $\{\lambda_j^a(\Omega,p)\}_{j\geq1}$ and $\{\lambda_j(\Omega,p)\}_{j\geq1}$ be the eigenvalues of eq:eige_equation_a and eq:eige_lapla, respectively. Then, for every $k \in \mathbb{N} \setminus \{0\}$,

Figures (1)

  • Figure 1: Definition of $P_a^\Omega$ and $\omega_a^\Omega$.

Theorems & Definitions (51)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Proposition 2.3
  • proof : Proof of Proposition \ref{['prop_spectral_stability']}
  • Remark 4.1
  • ...and 41 more