Exponential localization of hydrogen-like atoms in relativistic quantum electrodynamics

TL;DR

This paper proves exponential localization and semi-boundedness of the spectrum for relativistic models of hydrogen-like atoms in quantum electrodynamics, valid for all physical parameters below critical charges.

Contribution

It establishes the exponential localization and spectral semi-boundedness of two relativistic atom models in QED for arbitrary coupling constants and cut-offs, extending previous non-relativistic results.

Findings

No-pair operator is semi-bounded below.

Spectral subspaces below ionization threshold are exponentially localized.

Results hold for all values of fine-structure constant and UV cut-off within specified charge limits.

Abstract

We consider two different models of a hydrogenic atom in a quantized electromagnetic field that treat the electron relativistically. The first one is a no-pair model in the free picture, the second one is given by the semi-relativistic Pauli-Fierz Hamiltonian. We prove that the no-pair operator is semi-bounded below and that its spectral subspaces corresponding to energies below the ionization threshold are exponentially localized. Both results hold true, for arbitrary values of the fine-structure constant, $e^2$, and the ultra-violet cut-off, $\Lambda$, and for all nuclear charges less than the critical charge without radiation field, $Z_c=e^{-2}2/(2/\pi+\pi/2)$. We obtain similar results for the semi-relativistic Pauli-Fierz operator, again for all values of $e^2$ and $\Lambda$ and for nuclear charges less than $e^{-2}2/\pi$.