Spectral Analysis for Gaussian Quantum Markov Semigroups

Abstract

Let $(T_t)_{t\geq 0}$ be a Gaussian quantum Markov semigroup with a faithful normal invariant state $\rho$. For every $s\in[0,1]$, the $s$-embedding associated with $\rho$ induces a contraction semigroup $(T_t^{(s)})_{t\geq 0}$ on the Hilbert--Schmidt space $\mathcal B_2(\mathsf h)$; let $L^{(s)}$ denote its generator. Without assuming symmetry or quantum detailed balance, we determine the full spectra of $L^{(s)}$ and $L^{(s)\ast}$: they consist, respectively, of the non-negative integer combinations of the eigenvalues of the phase-space drift matrix and of their complex conjugates. We also diagonalize the self-adjoint closure of $L^{(s)\ast}+L^{(s)}$ and obtain an explicit formula for the spectral gap. Using quantum characteristic functions, we represent $T_t^{(s)}$, up to unitary equivalence, as a complex Gaussian integral operator and prove that its kernel is square-integrable for every $t>0$. Hence $T_t^{(s)}$ is a Hilbert--Schmidt operator on $\mathcal B_2(\mathsf h)$ for every $t>0$, and every induced semigroup is immediately compact. Consequently, $L^{(s)}$ and $L^{(s)\ast}$ have compact resolvent for all $s\in[0,1]$, and their point spectra exhaust their full spectra. These results provide a full quantum counterpart of classical Ornstein--Uhlenbeck spectral theory.