  The structure of time-dependent Gaussian solutions for the Kostin equation in
dissipative quantum mechanics is analyzed. Expanding the generic external
potential near the center of mass of the wave packet, one conclude that: the
center of mass follows the dynamics of a classical particle under the external
potential and a damping proportional to the velocity; the width of the wave
packet satisfy a non-conservative Pinney equation. An appropriate perturbation
theory is developed for the free particle case, solving the long standing
problem of finding analytic expressions for square integrable solutions of the
free Kostin equation. The associated Wigner function is also studied.
