Bound on the number of negative eigenvalues of two-dimensional Schr\"odinger operators on domains

TL;DR

This paper proves that for two-dimensional Schrödinger operators with Dirichlet boundary conditions, the upper bound on negative eigenvalues can be made independent of the domain size, refining Solomyak's result.

Contribution

It establishes that the constant in Solomyak's eigenvalue bound is domain-independent for Dirichlet boundary conditions in 2D.

Findings

The eigenvalue bound constant is independent of the domain size.

The bound applies to Schrödinger operators with Dirichlet boundary conditions.

The result refines previous domain-dependent bounds.

Abstract

A fundamental result of Solomyak says that the number of negative eigenvalues of a Schr\"odinger operator on a two-dimensional domain is bounded from above by a constant times a certain Orlicz norm of the potential. Here we show that in the case of Dirichlet boundary conditions the constant in this bound can be chosen independently of the domain.