On the eigenfunctions of no-pair operators in classical magnetic fields

TL;DR

This paper analyzes the spectral properties of a relativistic no-pair Hamiltonian for a hydrogenic atom in a classical magnetic field, establishing bounds, spectrum characteristics, and eigenfunction regularity and decay.

Contribution

It proves semi-boundedness of the Hamiltonian, characterizes its essential spectrum, and demonstrates exponential decay and smoothness of eigenfunctions under magnetic fields.

Findings

Hamiltonian is semi-bounded below for subcritical coupling constants.

Essential spectrum is [1, ∞) with infinitely many eigenvalues below 1.

Eigenfunctions are smooth away from the nucleus and decay exponentially.

Abstract

We consider a relativistic no-pair model of a hydrogenic atom in a classical, exterior magnetic field. First, we prove that the corresponding Hamiltonian is semi-bounded below, for all coupling constants less than or equal to the critical one known for the Brown-Ravenhall model, i.e., for vanishing magnetic fields. We give conditions ensuring that its essential spectrum equals [1,\infty) and that there exist infinitely many eigenvalues below 1. (The rest energy of the electron is 1 in our units.) Assuming that the magnetic vector potential is smooth and that all its partial derivatives increase subexponentially, we finally show that an eigenfunction corresponding to an eigenvalue \lambda<1 is smooth away from the nucleus and that its partial derivatives of any order decay pointwise exponentially with any rate a<(1-\lambda^2)^{1/2}, for \lambda\in[0,1), and a<1, for \lambda<0.