On localization of eigenfunctions of the magnetic Laplacian

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 $A:\Omega \to {\mathbb{R}}^d$, 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.