Liouville's equation as a Schrödinger equation

We show that every non-negative solution of Liouville's equation for an arbitrary (possibly non-Hamiltonian) dynamical system admits a factorization $\psi\psi^*$, where $\psi$ satisfies a Schrödinger equation of special form. The corresponding quantum system is obtained by Weyl quantization of a Hamiltonian system whose Hamiltonian is linear in the momenta. We discuss the structure of the spectrum of the special Schrödinger equation on a multidimensional torus and show that the eigenfunctions may have finite smoothness in the analytic case. Our generalized solutions of the Schrödinger equation are natural examples of non-selfadjoint extensions of Hermitian differential operators. We give conditions for the existence of a smooth invariant measure of a dynamical system. They are expressed in terms of stability conditions for the conjugate equations of variations.