Heat kernel estimates and spectral properties of a pseudorelativistic operator with magnetic field

TL;DR

This paper derives heat kernel estimates for a pseudorelativistic operator with magnetic field and analyzes its spectral properties, especially the essential spectrum of a two-electron system in combined Coulomb and magnetic fields.

Contribution

It provides new heat kernel estimates for a pseudorelativistic operator and determines the bottom of its essential spectrum in a two-particle Brown-Ravenhall model.

Findings

Derived heat kernel estimates for E_A operator

Determined the bottom of the essential spectrum for two-electron system

Spectral analysis restricted to central charge Z ≤ 86

Abstract

Based on the Mehler heat kernel of the Schroedinger operator for a free electron in a constant magnetic field an estimate for the kernel of E_A is derived, where E_A represents the kinetic energy of a Dirac electron within the pseudorelativistic no-pair Brown-Ravenhall model. This estimate is used to provide the bottom of the essential spectrum for the two-particle Brown-Ravenhall operator, describing the motion of the electrons in a central Coulomb field and a constant magnetic field, if the central charge is restricted to Z below or equal 86.