Spectral theory of no-pair Hamiltonians

TL;DR

This paper establishes a spectral analysis framework for no-pair Hamiltonians in atomic physics, proving an HVZ theorem, existence of infinitely many eigenvalues below the essential spectrum, and exponential localization of eigenfunctions.

Contribution

It extends spectral theory to include magnetic fields, non-local exchange potentials, and Dirac operators, providing new commutator estimates for these complex models.

Findings

Proved HVZ theorem for no-pair Hamiltonians with magnetic fields.

Showed existence of infinitely many eigenvalues below the essential spectrum.

Eigenfunctions are exponentially localized.

Abstract

We prove a HVZ theorem for a general class of no-pair Hamiltonians describing an atom or positively charged ion with several electrons in the presence of a classical external magnetic field. Moreover, we show that there exist infinitely many eigenvalues below the essential spectrum and that the corresponding eigenfunctions are exponentially localized. The novelty is that the electrostatic and magnetic vector potentials as well as a non-local exchange potential are included in the projection determining the model. As a main technical tool we derive various commutator estimates involving spectral projections of Dirac operators with external fields. Our results apply to all nuclear charges less than or equal to 137.