  Spatial localization of the electrons of an atom or molecule is studied in
models of non-relativistic matter coupled to quantized radiation. We give two
definitions of the ionization threshold. One in terms of spectral data of
cluster Hamiltonians, and one in terms of minimal energies of non-localized
states. We show that these two definitions agree, and that the electrons
described by a state with energy below the ionization threshold are localized
in a small neighborhood of the nuclei with a probability that approaches 1
exponentially fast with increasing radius of the neighborhood. The latter
result is derived from a new, general result on exponential decay tailored to
fit our problem, but applicable to many non-relativistic quantum systems
outside quantum electrodynamics as well.
