Time-Dependent Hamiltonians in Quantum Mechanics and Inflationary Cosmology

We revise the problem which appears in Quantum Mechanics when the Hamiltonian depends explicitly on time and provide a general setting to address such quantum systems. As a paradigmatic example we analyse the case of the damped harmonic oscillator (satisfying the Caldirola-Kanai equation) and extend the system to accomodate the ordinary time translation as a true symmetry (Bateman dual system). This general scheme applies in particular to the present problem of unitarity in Quantum Inflationary Cosmological Models.