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

TL;DR

This paper investigates the geometric shape of the level sets of the first eigenfunction of certain 2D Schrödinger operators with convex potentials, revealing how domain orientation and two key length scales influence these shapes.

Contribution

It introduces a novel analysis linking the shape of eigenfunction level sets to two specific length scales and domain orientation for a class of Schrödinger operators.

Findings

Identifies two length scales $L_1$ and $L_2$ that determine level set shapes.

Establishes bounds on the first eigenvalue using an associated ODE eigenvalue.

Describes how domain orientation affects eigenfunction level set geometry.

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.