On localization of eigenfunctions of the magnetic Laplacian
Let $Ω\subset \mathbb{R}^d$ and consider the magnetic Laplace operator given by $ H(A) = \left(- i\nabla - A(x)\right)^2$, where $A:Ω\rightarrow \mathbb{R}^d$, subject to Dirichlet eigenfunction. This operator can, for certain vector fields $A$, have eigenfunctions $H(A) ψ= λψ$ that are highly localized in a small region of $Ω$. The main goal of this paper is to show that if $|ψ|$ assumes its maximum in $x_0 \in Ω$, then $A$ behaves `almost' like a conservative vector field in a $1/\sqrtλ-$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.
math.AP↗