Quantum Electrodynamics of Atomic Resonances

Abstract

A simple model of an atom interacting with the quantized electromagnetic field is studied. The atom has a finite mass $m$, finitely many excited states and an electric dipole moment, $\vec{d}_0 = -λ_{0} \vec{d}$, where $\| d^{i}\| = 1,$ $ i=1,2,3,$ and $λ_0$ is proportional to the elementary electric charge. The interaction of the atom with the radiation field is described with the help of the Ritz Hamiltonian, $-\vec{d}_0\cdot \vec{E}$, where $\vec{E}$ is the electric field, cut off at large frequencies. A mathematical study of the Lamb shift, the decay channels and the life times of the excited states of the atom is presented. It is rigorously proven that these quantities are analytic functions of the momentum $\vec{p}$ of the atom and of the coupling constant $λ_0$, provided $|\vec{p}| < mc$ and $| \Im\vec{p} |$ and $| λ_{0} |$ are sufficiently small. The proof relies on a somewhat novel inductive construction involving a sequence of `smooth Feshbach-Schur maps' applied to a complex dilatation of the original Hamiltonian, which yields an algorithm for the calculation of resonance energies that converges super-exponentially fast.