The Shape of the Level Sets of the First Eigenfunction of a Class of Two Dimensional Schr\"odinger Operators

Abstract

We study the first Dirichlet eigenfunction of a class of Schr\"odinger operators with a convex potential V on a domain $\Omega$. We find two length scales $L_1$ and $L_2$, and an orientation of the domain $\Omega$, which determine the shape of the level sets of the eigenfunction. As an intermediate step, we also establish bounds on the first eigenvalue in terms of the first eigenvalue of an associated ordinary differential operator.