Exponential Decays of Steklov Eigenfunctions for the Magnetic Laplacian

TL;DR

This paper studies how the ground state of the Dirichlet-to-Neumann map for a magnetic Schrödinger operator decays exponentially in a bounded domain, especially near regions where the magnetic field vanishes.

Contribution

It demonstrates exponential decay of the ground state for large magnetic field strength, extending understanding of magnetic Schrödinger operators with finite type magnetic fields.

Findings

Ground state decays exponentially away from magnetic field vanishing regions.

Decay rate depends on the magnetic field strength parameter.

Results apply to magnetic fields of finite type.

Abstract

Consider the Dirichlet-to-Neumann map $\Lambda_\beta$ associated with the Schr\"odinger operator $(D+\beta \A)^2$ 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.