Trace formula of quantum Liouville operator

摘要

The spectral property of quantum Liouville operators (<((L) over cap)over cap>) is investigated by introducing its trace formula. It is shown that this trace formula coincides with the two-point level correlator R-2(-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 <((L) over cap)over cap>) 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. [References: 16]