Year: 2026
Type: preprint
Abstract: We characterize compactness of the Hilbert-Schmidt semigroups induced by finite-mode Gaussian quantum Markov semigroups with a faithful normal invariant state. A unitary second-quantization representation for every $ s $-embedding shows that compactness at one positive time, compact resolvent, drift stability, and uniqueness of the normal invariant state are equivalent. These conditions imply immediate trace-class membership for all embeddings. We derive exact trace-norm and Hilbert-Schmidt norm formulas, an embedding-independent trace identity, and the generator spectrum with algebraic multiplicities. Examples show that compactness does not require the existence of a spectral gap and that a spectral gap need not imply compactness when the invariant state is not unique.
Source: arXiv (Cornell University)
Author: Li, Zheng
Institution: Central South University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221539149