Singular-value decomposition of solution operators to model evolution equations

We consider evolution equations generated by quadratic operators admitting a decomposition in creation-annihilation operators without usual ellipticity-type hypotheses; this class includes hypocoercive model operators. We identify the singular value decomposition of their solution operators with the evolution generated by an operator of harmonic oscillator type, and in doing so derive exact characterizations of return to equilibrium and regularization for any complex time.