Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the magnetic Dirichlet to Neumann operator on the exterior of the disk -- diamagnetism, weak-magnetic field limit and flux effects

Helffer Bernard, Nicoleau Francois

TL;DR

This work analyzes the magnetic Dirichlet-to-Neumann map on the exterior of the unit disk under a combined constant magnetic field and Aharonov-Bohm flux, proving convergence to the interior map in the weak-field limit and establishing strong diamagnetism via large-field asymptotics. The authors derive explicit spectral data using confluent hypergeometric functions, characterize intersection points of spectral branches, and extend key interior-disk asymptotics to the exterior domain. They also incorporate AB flux effects, deriving precise weak- and strong-field behavior in the flux-extended problem and providing a framework applicable to general unbounded domains. The results advance understanding of magnetic boundary value problems in unbounded geometries and have implications for spectral geometry and diamagnetism theory.

Abstract

In this paper, we analyze the magnetic Dirichlet-to-Neumann operator (D-to-N map) $\check Λ(b,ν)$ on the exterior of the disk with respect to a magnetic potential $A_{b, ν}=A^b + A_ν$ where, for $b\in \mathbb R$ and $ν\in \mathbb R$, $A^b (x,y)= b\, (-y, x)$ and $A_ν(x,y)$ is the Aharonov-Bohm potential centered at the origin of flux $2πν$. First, we show that the limit of $\check Λ(b,ν)$ as $b\rightarrow 0$ is equal to the D-to-N map $\widehat Λ(ν)$ on the interior of the disk associated with the potential $A_ν(x,y)$. Secondly, we study the ground state energy of the D-to-N map $\check Λ(b,ν)$ and show that the strong diamagnetism property holds. Finally we slightly extend to the exterior case the asymptotic results obtained in the interior case for general domains.

On the magnetic Dirichlet to Neumann operator on the exterior of the disk -- diamagnetism, weak-magnetic field limit and flux effects

TL;DR

This work analyzes the magnetic Dirichlet-to-Neumann map on the exterior of the unit disk under a combined constant magnetic field and Aharonov-Bohm flux, proving convergence to the interior map in the weak-field limit and establishing strong diamagnetism via large-field asymptotics. The authors derive explicit spectral data using confluent hypergeometric functions, characterize intersection points of spectral branches, and extend key interior-disk asymptotics to the exterior domain. They also incorporate AB flux effects, deriving precise weak- and strong-field behavior in the flux-extended problem and providing a framework applicable to general unbounded domains. The results advance understanding of magnetic boundary value problems in unbounded geometries and have implications for spectral geometry and diamagnetism theory.

Abstract

In this paper, we analyze the magnetic Dirichlet-to-Neumann operator (D-to-N map) on the exterior of the disk with respect to a magnetic potential where, for and , and is the Aharonov-Bohm potential centered at the origin of flux . First, we show that the limit of as is equal to the D-to-N map on the interior of the disk associated with the potential . Secondly, we study the ground state energy of the D-to-N map and show that the strong diamagnetism property holds. Finally we slightly extend to the exterior case the asymptotic results obtained in the interior case for general domains.

Paper Structure

This paper contains 19 sections, 24 theorems, 160 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Generalities for the exterior problem with variable magnetic field
  3. Constant magnetic field in the disk--Reminder
  4. Magnetic Steklov eigenvalues on the exterior of the disk.
  5. Special functions
  6. Steklov eigenvalues
  7. Weak magnetic field limit of the exterior D-to-N map $\check\Lambda(b)$.
  8. Intersecting points for the case outside the disk.
  9. Characterization of the intersection points.
  10. The (G) formula.
  11. Computation of $\check \lambda'_n(z)$ and application to strong diamagnetism.
  12. Asymptotics of $\check z_n$.
  13. A complete asymptotic expansion.
  14. Applications.
  15. Additional flux effects.
  16. ...and 4 more sections

Key Result

Theorem 1.1

For any $b>0$ and any $\nu \in (-\frac{1}{2},\frac{1}{2}]$, $\check \Lambda(b,\nu)-\widehat{\Lambda}(\nu) \in {\cal{B}}(L^2(S^{1}))$. Moreover, we have as $b \to 0^+\,$:

Figures (6)

  • Figure 1: The Steklov eigenvalues $\lambda_n (b)$ (left) on the disk and the ground state energy $\lambda^{DN}(b)$ (right).
  • Figure 2: The magnetic Steklov eigenvalues $\check \lambda_n(b)$ (left) and the ground state energy $\check \lambda^{DN}(b)$ (right).
  • Figure 3: The magnetic Steklov eigenvalues $\check \lambda_0(b)$.
  • Figure 4: The graph of $U(-\frac{1}{2}, 5, x)$.
  • Figure 5: The magnetic Steklov eigenvalues $\check \lambda_n(b, \nu)$ for $\nu = \frac{1}{4}$ (left) and for $\nu=-\frac{1}{4}$ (right).
  • ...and 1 more figures

Theorems & Definitions (32)

  • Theorem 1.1
  • Proposition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 3.1
  • ...and 22 more