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

On Schroedinger Equation with Time-Dependent Quadratic Hamiltonian in $R^d$