  We study solutions to the Cauchy problem for the linear and nonlinear
Schroedinger equation with a quadratic Hamiltonian depending on time. For the
linear case the evolution operator can be expressed as an integral operator
with the explicit formula for the kernel. As a consequence, conditions for
local and global in time Strichartz estimates can be established. For the
nonlinear case we show local well-posedness. As a particular case we obtain
well-posedness for the damped harmonic nonlinear Schroedinger equation.
