Spectral Analysis for Gaussian Quantum Markov Semigroups

Abstract

Let $(T_t{)}_{t\ge 0}$ 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 $(T_t^{(s)}{)}_{t\ge 0}$ 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.

59 pages, 1 figure. Substantially revised and expanded version. We determine the full spectra for all s-embeddings, establish immediate compactness via quantum characteristic functions, and substantially strengthen the domain and compactness arguments