Smoothness and analyticity of perturbation expansions in QED

TL;DR

This paper proves that in non-relativistic QED, the atom's ground state energy and state are smoothly differentiable functions of the fine structure constant within a certain range, under specific assumptions.

Contribution

It establishes the smoothness and analyticity of the ground state and energy as functions of the fine structure constant in non-relativistic QED.

Findings

Ground state energy is k-times continuously differentiable.

Ground state is k-times continuously differentiable.

Results hold under specified ultraviolet cutoff and non-degenerate ground state assumptions.

Abstract

We consider the ground state of an atom in the framework of non-relativistic qed. We assume that the ultraviolet cutoff is of the order of the Rydberg energy and that the atomic Hamiltonian has a non-degenerate ground state. We show that the ground state energy and the ground state are k-times continuously differentiable functions of the fine structure constant and respectively the square root of the fine structure constant on some nonempty interval [0,c_k).