$L^p$ Estimates for Eigenfunctions of a Generalized Landau Magnetic Laplacian

Abstract

In this paper, we examine eigenfunctions of a generalized Landau Magnetic Laplacian that models the physics of an electron confined to a plane in a magnetic field orthogonal to the plane. This operator has an infinite dimensional null space and, at least in the model case, has infinite dimensional eigenspaces with eigenvalues which are essentially the same as the eigenvalues of the Hermite operator. We demonstrate that, under fairly general assumptions on the potential function of the magnetic field, the $L^\infty$ norm of these eigenfunctions is bounded by their $L^2$ norm independently of the associated eigenvalue. We furthermore demonstrate an improvement in the $L^6$ norm of these eigenfunctions. The method we use comes from semiclassical analysis and is inspired by the work of Koch, Tataru, and Zworski that applies locally. In our case, we use a new conjugation argument to demonstrate the result over all of $\mathbb{R}^2$.