On Localization of Eigenfunctions of The Magnetic Laplacian

Let Ω ⊂ ℝ^d and consider the magnetic Laplace operator given by H(A) = (–i∇ – A(x))², where A : Ω → ℝ^d, subject to Dirichlet boundary conditions. For certain vector fields A, this operator can 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 at x₀ ∈ Ω, then A behaves 'almost' like a conservative vector field in a $1/\sqrt{\lambda }$-neighborhood of x₀ in a precise sense. In particular, we expect localization in regions where |curl A| is small. The result is illustrated with numerical examples.