On negative eigenvalues of two-dimensional Schroedinger operators with singular potentials

TL;DR

This paper provides upper bounds on the number of negative eigenvalues for 2D Schrödinger operators with singular potentials supported on Ahlfors regular measures, improving known estimates especially for measures supported on Lipschitz curves.

Contribution

It introduces new upper estimates for negative eigenvalues of 2D Schrödinger operators with singular potentials supported on arbitrary Ahlfors regular measures, extending and strengthening previous results.

Findings

Derived upper bounds involving logarithmic weighted integrals of the potential.

Established stronger estimates for potentials supported on Lipschitz curves.

Applicable to potentials generated by measures of arbitrary dimension within (0,2].

Abstract

We present upper estimates for the number of negative eigenvalues of two-dimensional Schroedinger operators with potentials generated by Ahlfors regular measures of arbitrary dimension $\alpha\in (0, 2]$.The estimates are given in terms of the integrals of the potential with a logarithmic weight and of its L $\log$ L type Orlicz norms. In the case $\alpha = 1$, our estimates are stronger than the known ones about Schroedinger operators with potentials supported by Lipschitz curves.