Trace-class spectra of irreducible Gaussian quantum Markov semigroups

Abstract

We determine the spectrum of the predual generator of an irreducible Gaussian quantum Markov semigroup on the trace-class operators over a finite-mode bosonic Fock space in the stable, strictly unstable, and periodic critical drift regimes. For stable drift, irreducibility is equivalent to the existence of a unique faithful normal invariant state. The spectrum and approximate point spectrum are the closed left half-plane, whereas the point spectrum is the open left half-plane together with zero. Thus the polynomial eigenvalues generated by the drift matrix do not exhaust the trace-class point spectrum. If the drift has an eigenvalue with strictly positive real part, the spectrum is again the closed left half-plane, but the point spectrum is empty and the open left half-plane together with zero belongs to the residual spectrum. For periodic critical drift, we obtain an explicit spectral formula that includes the displacement parameter and yields horizontal half-lines or parabolic regions. The proofs use characteristic functions, Gaussian diffusion semigroups, and bounded eigenoperators of the dual semigroup. The nonperiodic critical drift case remains open.