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Asymptotics for the magnetic Dirichlet-to-Neumann eigenvalues in general domains

Bernard Helffer, Ayman Kachmar, Francois Nicoleau

TL;DR

This work establishes rigorous asymptotics for the ground-state energy of the magnetic Dirichlet-to-Neumann operator on general 2D and 3D domains as the magnetic field strength grows. The authors derive a leading order $O(b^{1/2})$ term determined by the boundary magnetic field magnitude, with 2D refinements showing curvature-driven corrections and eigenvalue splitting via a Robin-model comparison; in 3D, the asymptotics reduce to a boundary-oriented infimum involving orientation-dependent 2D models. They also connect these DN asymptotics to the magnetic Robin Laplacian and analyze the weak-field regime, obtaining a quadratic-in-$b$ expansion governed by a boundary integral of a gauge-canonical potential. Collectively, the results generalize prior disk/half-plane analyses to general domains and illuminate how boundary geometry and magnetic orientation control spectral behavior in strong-field regimes.

Abstract

Inspired by a paper by T. Chakradhar, K. Gittins, G. Habib and N. Peyerimhoff, we analyze their conjecture that the ground state energy of the magnetic Dirichlet-to-Neumann operator tends to infinity as the magnetic field tends to infinity. More precisely, we prove refined conjectures for general two dimensional domains, based on the analysis in the case of the half-plane and the disk by two of us (B.H. and F.N.). We also extend our analysis to the three dimensional case, and explore a connection with the eigenvalue asymptotics of the magnetic Robin Laplacian.

Asymptotics for the magnetic Dirichlet-to-Neumann eigenvalues in general domains

TL;DR

This work establishes rigorous asymptotics for the ground-state energy of the magnetic Dirichlet-to-Neumann operator on general 2D and 3D domains as the magnetic field strength grows. The authors derive a leading order term determined by the boundary magnetic field magnitude, with 2D refinements showing curvature-driven corrections and eigenvalue splitting via a Robin-model comparison; in 3D, the asymptotics reduce to a boundary-oriented infimum involving orientation-dependent 2D models. They also connect these DN asymptotics to the magnetic Robin Laplacian and analyze the weak-field regime, obtaining a quadratic-in- expansion governed by a boundary integral of a gauge-canonical potential. Collectively, the results generalize prior disk/half-plane analyses to general domains and illuminate how boundary geometry and magnetic orientation control spectral behavior in strong-field regimes.

Abstract

Inspired by a paper by T. Chakradhar, K. Gittins, G. Habib and N. Peyerimhoff, we analyze their conjecture that the ground state energy of the magnetic Dirichlet-to-Neumann operator tends to infinity as the magnetic field tends to infinity. More precisely, we prove refined conjectures for general two dimensional domains, based on the analysis in the case of the half-plane and the disk by two of us (B.H. and F.N.). We also extend our analysis to the three dimensional case, and explore a connection with the eigenvalue asymptotics of the magnetic Robin Laplacian.
Paper Structure (38 sections, 31 theorems, 268 equations, 1 figure)

This paper contains 38 sections, 31 theorems, 268 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The magnetic Dirichlet to Neumann operator.
  3. Planar domains
  4. Three dimensional case
  5. The magnetic Robin Laplacian
  6. Weak field limit
  7. Organization
  8. Preliminaries
  9. The half-plane
  10. Harmonic oscillator with Robin condition
  11. D-to-N on the half-axis
  12. A useful identity
  13. Gauge invariance
  14. Parallel coordinates in two dimensions
  15. Upper bounds in two dimensional domains
  16. ...and 23 more sections

Key Result

Theorem 1.1

Let $\Omega$ be a regular domain in $\mathbb R^2$ and $A$ be a magnetic potential with constant magnetic field with norm $1$. Then the ground state energy of the D-to-N map $\Lambda_{bA}$ satisfies where $-\alpha$ is the unique negative zero of the parabolic cylindrical function $D_{1/2}(z)$.

Figures (1)

  • Figure 1: Graph of the function $\Theta(\gamma)$.

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 45 more