Effective Hamiltonians for atoms in very strong magnetic fields

TL;DR

This paper introduces three effective Hamiltonians to approximate atoms in very strong magnetic fields, simplifying the Coulomb interaction while retaining key spectral properties, aiding spectral analysis.

Contribution

The paper proposes three novel effective Hamiltonians for atoms in strong magnetic fields, replacing Coulomb interactions with B-dependent potentials, and discusses their solvability and spectral analysis applications.

Findings

Two Hamiltonians are solvable for one electron.

The Hamiltonians approximate the spectral bottom of atoms in strong fields.

Potential for analyzing spectral properties in extreme magnetic regimes.

Abstract

We propose three effective Hamiltonians which approximate atoms in very strong homogeneous magnetic fields $B$ modelled by the Pauli Hamiltonian, with fixed total angular momentum with respect to magnetic field axis. All three Hamiltonians describe $N$ electrons and a fixed nucleus where the Coulomb interaction has been replaced by $B$-dependent one-dimensional effective (vector valued) potentials but without magnetic field. Two of them are solvable in at least the one electron case. We briefly sketch how these Hamiltonians can be used to analyse the bottom of the spectrum of such atoms.