Quasi-classical ground states in non-relativistic QED and related models

We will consider in this talk a non-relativistic particle bound by an external potential and coupled to a quantized radiation field. This physical system is mathematically described by a Pauli-Fierz Hamiltonian. We will study the energy functional of product states of the form $u\otimes \Psi_f$, where $u$ is a normalized state for the non-relativistic particle and $\Psi_f$ is a coherent state in Fock space for the field. This gives the energy of a Klein-Gordon-Schrödinger system in the case of a spinless particle linearly coupled to a scalar field, or the energy of a Maxwell-Schrödinger system in the case of an electron coupled to the photon field. In both cases, we will discuss results concerning the existence and uniqueness of a ground state, under general conditions on the external potential and the coupling form factor. In particular, neither an ultraviolet cutoff nor an infrared cutoff needs to be imposed. We will
also discuss the convergence in the ultraviolet limit and the second-order asymptotic expansion in the coupling constant of the ground state energy.

This is joint work with J. Payet and S. Breteaux.