  This paper is devoted to investigation of the quantum
Friedman-Robertson-Walker universe with matter satisfying the equation of state
$p=w\rho$, where $w$ is an almost arbitrary constant. The procedure starts with
a reduced Lagrangian, which describes the system in a gauge fixed, so that the
evolution parameter corresponds to the cosmological time. Then we construct the
phase space, which is believed to correspond to the reduced phase space
consisting of Dirac's observables. The physically relevant quantities are
mapped into operators. We show that the operators have self-adjoint
realizations and that there exist quantum states for which the evolution across
singularity is well-defined.
