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The Magnetic Laplacian with a Higher-order Vanishing Magnetic Field in a Bounded Domain

Zhongwei Shen

TL;DR

This work presents a unified semi-classical analysis of the magnetic Laplacian in bounded domains under highly vanishing magnetic fields. By introducing a vanishing-order framework for the magnetic field and coupling it with local model problems built from Taylor polynomials, the authors derive the leading $eta$-scalings for the Dirichlet, Neumann, and Dirichlet-to-Neumann spectra, and provide remainder estimates under additional structural assumptions on invariant subspaces. The key contributions include a set of sharp upper and lower bounds, local asymptotic expansions with explicit error terms, and detailed analysis for polynomial fields, culminating in concrete examples that recover classical non-vanishing and vanishing scenarios as well as the two-dimensional first-order vanishing case. The results offer a coherent, geometry-informed method to understand how vanishing magnetic fields govern ground-state energies in varied boundary conditions, with potential implications for superconductivity and related quantum systems.

Abstract

This paper is concerned with spectrum properties of the magnetic Laplacian with a higher-order vanishing magnetic field in a bounded domain. We study the asymptotic behaviors of ground state energies for the Dirichlet Laplacian, the Neumann Laplacian, and the Dirichlet-to-Neumann operator, as the field strength parameter $β$ goes to infinite. Assume that the magnetic field does not vanish to infinite order, we establish the leading orders of $β$. We also obtain the first terms in the asymptotic expansions with remainder estimates under additional assumptions on an invariant subspace for a Taylor polynomial of the magnetic field. Our aim is to provide a unified approach to all three cases.

The Magnetic Laplacian with a Higher-order Vanishing Magnetic Field in a Bounded Domain

TL;DR

This work presents a unified semi-classical analysis of the magnetic Laplacian in bounded domains under highly vanishing magnetic fields. By introducing a vanishing-order framework for the magnetic field and coupling it with local model problems built from Taylor polynomials, the authors derive the leading -scalings for the Dirichlet, Neumann, and Dirichlet-to-Neumann spectra, and provide remainder estimates under additional structural assumptions on invariant subspaces. The key contributions include a set of sharp upper and lower bounds, local asymptotic expansions with explicit error terms, and detailed analysis for polynomial fields, culminating in concrete examples that recover classical non-vanishing and vanishing scenarios as well as the two-dimensional first-order vanishing case. The results offer a coherent, geometry-informed method to understand how vanishing magnetic fields govern ground-state energies in varied boundary conditions, with potential implications for superconductivity and related quantum systems.

Abstract

This paper is concerned with spectrum properties of the magnetic Laplacian with a higher-order vanishing magnetic field in a bounded domain. We study the asymptotic behaviors of ground state energies for the Dirichlet Laplacian, the Neumann Laplacian, and the Dirichlet-to-Neumann operator, as the field strength parameter goes to infinite. Assume that the magnetic field does not vanish to infinite order, we establish the leading orders of . We also obtain the first terms in the asymptotic expansions with remainder estimates under additional assumptions on an invariant subspace for a Taylor polynomial of the magnetic field. Our aim is to provide a unified approach to all three cases.
Paper Structure (12 sections, 45 theorems, 328 equations)

This paper contains 12 sections, 45 theorems, 328 equations.

Table of Contents

  1. Introduction
  2. Upper bounds, part I
  3. Operator lower bounds
  4. Proofs of Theorems \ref{['main-thm-1']} and \ref{['main-thm-2']}
  5. Polynomial magnetic fields
  6. Local asymptotic expansions
  7. Upper bounds, part II
  8. Asymptotic expansions
  9. Examples
  10. The non-vanishing case
  11. The case of discrete wells
  12. The first-order vanishing for $d=2$

Key Result

Theorem 1.1

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^d, d\ge 2$. Suppose that $\mathbf{A} \in C^\infty(\overline{\Omega}; \mathbb{R}^d)$ and $|\mathbf{B}|$ does not vanish to infinite order at any point in $\overline{\Omega}$. Let $\kappa_*\ge 0$ be defined by kappa. Then for $\beta>C$, where $C, c>0$ depend only on $d$, $\kappa_*$, $\Omega$ and $(C_0, c_0)$ in main-c-1.

Theorems & Definitions (108)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • ...and 98 more