Existence of atoms and molecules in non-relativistic quantum electrodynamics

We show that the Hamiltonian describing N nonrelativistic electrons with spin, interacting with the quantized radiation field and several fixed nuclei with total charge Z, has a ground state when N < Z + 1. The result holds for any value of the fine structure constant α and for any value of the ultraviolet cutoff A on the radiation field. There is no infrared cutoff. The basic mathematical ingredient in our proof is a novel localization of the electromagnetic field in such a way that the errors in the energy are of smaller order than 1/L, where L is the localization radius.