Exponential Decays of Steklov Eigenfunctions for the Magnetic Laplacian

Abstract

Consider the Dirichlet-to-Neumann map ${\Lambda }_{\beta }$ associated with the Schrödinger operator $(D+β\A)^2$ with a magnetic potential in a bounded Lipschitz domain $\Omega$, where $\beta >1$ is the field strength parameter. Assume that the magnetic field $\B=\nabla \times \A$ is of finite type. We show that if $\beta >{\beta }_0$, the ground state for ${\Lambda }_{\beta }$ decays exponentially away from a neighborhood of the subset of $\partial \Omega$, on which $\B$ vanishes to the maximal order.

15 pages. Comments are welcome