  In this article we investigate from the point of view of spectral theory the
problem of relaxation to thermodynamical equilibrium of a quantum harmonic
oscillator interacting with a radiation field. Our starting point is a system
of infinitely many Pauli master equations governing the time evolution of the
occupation probabilities of the available quantum states. The system we
consider is derived from the evolution equation for the reduced density
operator obtained after the initial interaction of the oscillator with the
radiation field, the latter acting as a heat bath. We provide a complete
spectral analysis of the infinitesimal generator of the equations, showing
thereby that it generates an infinite-dimensional system of hyperbolic type.
This implies that every global solution to the equations converge exponentially
rapidly toward the corresponding Gibbs equilibrium state. We also provide a
complete spectral analysis of a linear pencil naturally associated with the
Pauli equations. All of our considerations revolve around the notion of
compactness, more specifically around the existence of a new compact embedding
result involving a space of sequences of Sobolev type into a weighted space of
square-integrable summable sequences.
