100 years of Weyl's law

TL;DR

This paper reviews a century of Weyl's law, focusing on eigenvalue asymptotics for differential operators, including advanced topics like correction terms, non-Weyl asymptotics, and applications to quantum physics.

Contribution

It provides a comprehensive overview of Weyl asymptotics, extending classical results to include correction terms, non-Weyl parts, and applications to Schrödinger and Dirac operators under magnetic fields.

Findings

Analysis of sharp eigenvalue asymptotics

Extensions to Schrödinger and Dirac operators with magnetic fields

Applications to ground state energy estimates for atoms and molecules

Abstract

We discuss the asymptotics of the eigenvalue counting function for partial differential operators and related expressions paying the most attention to the sharp asymptotics. We consider Weyl asymptotics, asymptotics with Weyl principal parts and correction terms and asymptotics with non-Weyl principal parts. Semiclassical microlocal analysis, propagation of singularities and related dynamics play crucial role. We start from the general theory, then consider Schr\"odinger and Dirac operators with the strong magnetic field and, finally, applications to the asymptotics of the ground state energy of heavy atoms and molecules with or without a magnetic field.