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Exponential Decays of Steklov Eigenfunctions for the Magnetic Laplacian

Zhongwei Shen

TL;DR

This work analyzes the exponential decay of Steklov-type eigenfunctions for the magnetic Dirichlet-to-Neumann map associated with the magnetic Laplacian $H(β\mathbf{A})=(D+β\mathbf{A})^2$ on a bounded Lipschitz domain. By constructing an Agmon distance $d_β$ weighted by the magnetic field and exploiting the finite-type condition on $\mathbf{B}$, the authors establish $L^2$ and pointwise decay estimates for eigenfunctions when the field strength $β$ is large. The main finding is that ground-state eigenfunctions decay exponentially away from the boundary subset where $\mathbf{B}$ vanishes to the maximal order, with quantitative bounds expressed through the magnetic wells $W_β(t)$ and the distance $d_β$. The results also cover different vanishing scenarios, including non-vanishing and higher-order vanishing of the magnetic field, and provide explicit implications for the localization near boundary magnetic wells.

Abstract

Consider the Dirichlet-to-Neumann map $Λ_β$ associated with the Schrödinger operator $(D+β\A)^2$ with a magnetic potential in a bounded Lipschitz domain $Ω$, where $β>1$ is the field strength parameter. Assume that the magnetic field $\B=\nabla \times \A$ is of finite type. We show that if $β>β_0$, the ground state for $Λ_β$ decays exponentially away from a neighborhood of the subset of $\partialΩ$, on which $\B$ vanishes to the maximal order.

Exponential Decays of Steklov Eigenfunctions for the Magnetic Laplacian

TL;DR

This work analyzes the exponential decay of Steklov-type eigenfunctions for the magnetic Dirichlet-to-Neumann map associated with the magnetic Laplacian on a bounded Lipschitz domain. By constructing an Agmon distance weighted by the magnetic field and exploiting the finite-type condition on , the authors establish and pointwise decay estimates for eigenfunctions when the field strength is large. The main finding is that ground-state eigenfunctions decay exponentially away from the boundary subset where vanishes to the maximal order, with quantitative bounds expressed through the magnetic wells and the distance . The results also cover different vanishing scenarios, including non-vanishing and higher-order vanishing of the magnetic field, and provide explicit implications for the localization near boundary magnetic wells.

Abstract

Consider the Dirichlet-to-Neumann map associated with the Schrödinger operator with a magnetic potential in a bounded Lipschitz domain , where is the field strength parameter. Assume that the magnetic field is of finite type. We show that if , the ground state for decays exponentially away from a neighborhood of the subset of , on which vanishes to the maximal order.

Paper Structure

This paper contains 5 sections, 12 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. An Agmon distance function
  3. $L^2$ bounds
  4. Pointwise bounds
  5. Some examples

Key Result

Theorem 1.1

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^d$, $d\ge 2$ and $\mathbf{A}\in C^\infty(\overline{\Omega}, \mathbb{R}^d)$. Assume that $\mathbf{B}$ is of finite type on $\overline{\Omega}$. Let $u\in H^1(\Omega; \mathbb{C})$ be a weak solution of for some $\lambda>0$. Then, if $\beta > \beta_0$, for any $x\in \overline{\Omega}$. The constants $C>1, \beta_0>1, \varepsilon >0$ depend on

Theorems & Definitions (25)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 15 more