  In the framework of the Lindblad theory for open quantum systems the damping
of the harmonic oscillator is studied. A generalization of the fundamental
constraints on quantum mechanical diffusion coefficients which appear in the
master equation for the damped quantum oscillator is presented; the
Schr\"odinger and Heisenberg representations of the Lindblad equation are given
explicitly. On the basis of these representations it is shown that various
master equations for the damped quantum oscillator used in the literature are
particular cases of the Lindblad equation and that the majority of these
equations are not satisfying the constraints on quantum mechanical diffusion
coefficients. Analytical expressions for the first two moments of coordinate
and momentum are also obtained by using the characteristic function of the
Lindblad master equation. The master equation is transformed into Fokker-Planck
equations for quasiprobability distributions. A comparative study is made for
the Glauber $P$ representation, the antinormal ordering $Q$ representation and
the Wigner $W$ representation. It is proven that the variances for the damped
harmonic oscillator found with these representations are the same. By solving
the Fokker-Planck equations in the steady state, it is shown that the
quasiprobability distributions are two-dimensional Gaussians with widths
determined by the diffusion coefficients. The density matrix is represented via
a generating function, which is obtained by solving a time-dependent linear
partial differential equation derived from the master equation. Illustrative
examples for specific initial conditions of the density matrix are provided.
