Weyl's law for Neumann Schr\"{o}dinger operators on H\"{o}lder domains

TL;DR

This paper reviews recent advances in understanding the semiclassical spectral distribution of Neumann Schrödinger operators on Hölder domains, emphasizing conditions for Weyl's law validity and boundary bound state control.

Contribution

It introduces new techniques for estimating bound states near boundaries, extending Weyl's law applicability to more general domains with Neumann conditions.

Findings

Weyl's law holds under specific potential conditions.

Boundary bound states can be effectively controlled.

Universal estimates on the number of bound states are established.

Abstract

We review recent results on the semiclassical behaviour of Schr\"{o}dinger operators with Neumann boundary conditions. In this setting, the validity of Weyl's law requires additional conditions on the potential. We will explain the techniques needed to control the number of bound states near the boundary, thus leading to universal estimates on the number of bound states.