Exponential Decays of Steklov Eigenfunctions for the Magnetic Laplacian

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