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Eigenvalues of the magnetic Dirichlet Laplacian with constant magnetic field on disks in the strong field limit

Matthias Baur, Timo Weidl

TL;DR

The paper analyzes the eigenvalues λ_n(Ω,B) of the magnetic Dirichlet Laplacian with a constant field B on planar domains of finite measure, focusing on disks and the strong-field regime. For disks, all eigenvalue branches λ_{m,l}(B) are shown to satisfy λ_{m,l}(B)/B→l+|l|+1+2(m−1) as B→∞, with an explicit exponentially small remainder given by the confluent hypergeometric structure via M(a,b,z). It then proves that the magnetic Pólya-type bound is violated for large B with a universal excess factor 1/2, and this sharpness extends from disks to finite unions and general domains using domain-inclusion arguments. The results combine explicit disk analyses (root structure of M(a,b,z)) with domain-approximation techniques to derive universal spectral bounds in strong magnetic fields, with implications for Weyl-type asymptotics and Riesz means in magnetized settings.

Abstract

We consider the magnetic Dirichlet Laplacian with constant magnetic field on domains of finite measure. First, in the case of a disk, we prove that the eigenvalue branches with respect to the field strength behave asymptotically linear with an exponentially small remainder term as the field strength goes to infinity. We compute the asymptotic expression for this remainder term. Second, we show that for sufficiently large magnetic field strengths, the spectral bound corresponding to the Pólya conjecture for the non-magnetic Dirichlet Laplacian is violated up to a sharp excess factor which is independent of the domain.

Eigenvalues of the magnetic Dirichlet Laplacian with constant magnetic field on disks in the strong field limit

TL;DR

The paper analyzes the eigenvalues λ_n(Ω,B) of the magnetic Dirichlet Laplacian with a constant field B on planar domains of finite measure, focusing on disks and the strong-field regime. For disks, all eigenvalue branches λ_{m,l}(B) are shown to satisfy λ_{m,l}(B)/B→l+|l|+1+2(m−1) as B→∞, with an explicit exponentially small remainder given by the confluent hypergeometric structure via M(a,b,z). It then proves that the magnetic Pólya-type bound is violated for large B with a universal excess factor 1/2, and this sharpness extends from disks to finite unions and general domains using domain-inclusion arguments. The results combine explicit disk analyses (root structure of M(a,b,z)) with domain-approximation techniques to derive universal spectral bounds in strong magnetic fields, with implications for Weyl-type asymptotics and Riesz means in magnetized settings.

Abstract

We consider the magnetic Dirichlet Laplacian with constant magnetic field on domains of finite measure. First, in the case of a disk, we prove that the eigenvalue branches with respect to the field strength behave asymptotically linear with an exponentially small remainder term as the field strength goes to infinity. We compute the asymptotic expression for this remainder term. Second, we show that for sufficiently large magnetic field strengths, the spectral bound corresponding to the Pólya conjecture for the non-magnetic Dirichlet Laplacian is violated up to a sharp excess factor which is independent of the domain.
Paper Structure (11 sections, 8 theorems, 133 equations, 2 figures)

This paper contains 11 sections, 8 theorems, 133 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and statement of results
  3. Explicit solution of the eigenvalue problem on disks
  4. Asymptotics for eigenvalue branches of the disk
  5. Spectral inequalities and sharpness
  6. Proof of Theorem \ref{['thm:eigenvalue_asymptotics_disk']}
  7. Proof of Theorem \ref{['thm:magnetic_polya_general_set']} and Corollary \ref{['cor:magnetic_disk_trace']}
  8. Proof of Theorem \ref{['thm:magnetic_polya_general_set']} for disks
  9. Proof of Theorem \ref{['thm:magnetic_polya_general_set']} for disjoint unions of finitely many disks
  10. Proof of Theorem \ref{['thm:magnetic_polya_general_set']} for general domains
  11. Proof of Corollary \ref{['cor:magnetic_disk_trace']}

Key Result

Theorem 2.1

For any fixed $m\in\mathbb{N}$, $l\in\mathbb{Z}$ and $R>0$ holds and as $B\rightarrow +\infty$.

Figures (2)

  • Figure 1: Eigenvalues of the magnetic Dirichlet Laplacian on a disk of radius of $R=1/\sqrt{\pi}$ (and hence area $|D_R|=1$). Color of the curves fades from red to blue as $n$ increases. (a) Low eigenvalues $\lambda_n(D_R,B)$ over the field strength $B$. The dashed lines indicate the lines $\lambda = B$, $3B$ and $5B$. (b) The first 500 eigenvalues divided by $4\pi n / |D_R|$ plotted over the field strength $B$. The value $1.0$ associated with the lower bound from the non-magnetic Pólya conjecture is crossed for the first time at $B_{\mathrm{crit}} \approx 110.335$ by $\lambda_{11}(D_R, B)$. This point is marked by a black dot and a dashed line.
  • Figure 2: The first few roots $a_m(z)$ and $z_m(a)$ for $b=1$.

Theorems & Definitions (14)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Lemma 3.1
  • proof
  • Corollary 4.1
  • proof
  • Corollary 4.2
  • proof
  • Lemma 4.3
  • ...and 4 more