Quantum damped oscillator in a magnetic field

The problem of the quantal description of multidimensional nonhamiltonian systems with linear classical equations of motion is solved in the framework of the Fokker-Planck equation for the Wigner function. The conditions ensuring the preservation of the normalization, hermiticity, and positive definiteness of the density operator are established. The two-dimensional damped isotropic quantal oscillator in a uniform magnetic field is considered in detail, and the single analytic description suitable for arbitrary values of the damping coefficient, the frequencies and temperature is obtained.