Convergent expansions in non-relativistic QED: Analyticity of the ground state

TL;DR

This paper proves that the ground state and energy of an atom in non-relativistic QED are analytic functions of the coupling constant, with convergent power series expansions in the fine structure constant under certain conditions.

Contribution

It establishes the analyticity of the ground state and energy in non-relativistic QED and links the expansion coefficients to the Raleigh-Schroedinger series, providing rigorous mathematical results.

Findings

Ground state and energy are analytic functions of the coupling constant.

Expansion coefficients match the Raleigh-Schroedinger series.

Convergent power series in the fine structure constant in a specific scaling limit.

Abstract

We consider the ground state of an atom in the framework of non-relativistic qed. We show that the ground state as well as the ground state energy are analytic functions of the coupling constant which couples to the vector potential, under the assumption that the atomic Hamiltonian has a non-degenerate ground state. Moreover, we show that the corresponding expansion coefficients are precisely the coefficients of the associated Raleigh-Schroedinger series. As a corollary we obtain that in a scaling limit where the ultraviolet cutoff is of the order of the Rydberg energy the ground state and the ground state energy have convergent power series expansions in the fine structure constant $\alpha$, with $\alpha$ dependent coefficients which are finite for $\alpha \geq 0$.