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

2014/11/26 by Thomas Beck, Beck, Thomas
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #math.AP

paper · pdf · doi:10.48550/arxiv.1411.7353

56 pages, 3 figures

arxiv created 2014/11/26 · openalex publication_date 2014/11/26 · arxiv updated 2014/11/27 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

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.

Related

• Uniform level set estimates for ground state eigenfunctions
• Estimates on the Lower Bound of the First Gap
• Quantitative inequality for the eigenvalue of a Schrödinger operator in the ball
• Extremal properties of the first eigenvalue of Schrödinger-type operators
• Inequalities between Neumann and Dirichlet eigenvalues of Schrödinger operators