On localization of eigenfunctions of the magnetic Laplacian

TL;DR

This paper investigates how eigenfunctions of the magnetic Laplacian localize in regions where the magnetic field's curl is small, showing that the vector potential behaves nearly conservatively near maximum points of eigenfunctions.

Contribution

It establishes a relationship between eigenfunction localization and the near-conservativeness of the magnetic vector potential in high-energy regimes.

Findings

Eigenfunctions localize where the curl of A is small.

A behaves almost like a conservative field near eigenfunction maxima.

Numerical examples support the theoretical results.

Abstract

Let $\Omega \subset \mathbb{R}^d$ and consider the magnetic Laplace operator given by $ H(A) = \left(- i\nabla - A(x)\right)^2$, where $A:\Omega \rightarrow \mathbb{R}^d$, subject to Dirichlet eigenfunction. This operator can, for certain vector fields $A$, have eigenfunctions $H(A) \psi = \lambda \psi$ that are highly localized in a small region of $\Omega$. The main goal of this paper is to show that if $|\psi|$ assumes its maximum in $x_0 \in \Omega$, then $A$ behaves `almost' like a conservative vector field in a $1/\sqrt{\lambda}-$neighborhood of $x_0$ in a precise sense: we expect localization in regions where $\left|\mbox{curl} A \right|$ is small. The result is illustrated with numerical examples.