The Shape of the Level Sets of the First Eigenfunction of a Class of Two Dimensional Schrödinger Operators

We study the first Dirichlet eigenfunction of a class of Schrödinger operators with a convex potential V on a domain Ω. We find two length scales L1 and L2, and an orientation of the domain Ω, 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.