On the eigenvalues of operators with gaps. Application to Dirac operators

TL;DR

This paper develops a min-max method to characterize eigenvalues in spectral gaps of self-adjoint operators and applies it to Dirac operators with Coulomb-like potentials, achieving optimal results.

Contribution

It introduces a general abstract theorem for eigenvalues in spectral gaps and applies it specifically to Dirac operators with Coulomb potentials, demonstrating optimality.

Findings

Established a min-max characterization for spectral gap eigenvalues

Applied the theory to Dirac operators with Coulomb-like potentials

Achieved optimal results for Coulomb potential case

Abstract

This paper is devoted to a general min-max characterization of the eigenvalues in a gap of the essential spectrum of a self-adjoint unbounded operator. We prove an abstract theorem, then we apply it to the case of Dirac operators with a Coulomb-like potential. The result is optimal for the Coulomb potential. An erratum is appended at the end of the tex.