On localization of eigenfunctions of the magnetic Laplacian

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ and consider the magnetic Laplace operator given by $H(A)={(-i\mathrm{\nabla }-A(x))}^2$, where $A:\mathrm{\Omega }\to {\mathbb{R}}^d$, 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 $\mathrm{\Omega }$. The main goal of this paper is to show that if $\mathrm{\mid }\psi \mathrm{\mid }$ assumes its maximum in $x_0\in \mathrm{\Omega }$, then $A$ behaves `almost’ like a conservative vector field in a $1\mathrm{/}\sqrt{\lambda }-$neighborhood of $x_0$ in a precise sense: we expect localization in regions where $\mid curlA\mid$ is small. The result is illustrated with numerical examples.