Trace formula of quantum Liouville operator

Abstract

The spectral property of quantum Liouville operators $(\stackrel{^}{\stackrel{^}{L}})$ is investigated by introducing its trace formula. It is shown that this trace formula coincides with the two-point level correlator ${R}_{2}(\ensuremath{-}is)$ except some coefficients for quantized maps on a torus. Using semiclassical theory, for quantized chaotic systems, this enables us to write the trace formula (i.e., the spectrum of $\stackrel{^}{\stackrel{^}{L}}$) in terms of Pollicott-Ruelle resonances. Consequently, it is shown that the decay rates of density matrix is just semiclassically determined by the Pollicott-Ruelle resonances for the classical counterpart.