Quantum systems with time-dependent harmonic part and the Morse index

A simple relation between two quantum systems with a time-dependent, respectively, time-independent, harmonic part is established. Using this we give a computation, valid for all times, of the Green’s functions of the time-dependent harmonic oscillator with and without a perturbation of the type g/x2. The asymptotic expansion of the wave function in powers of Planck’s constant is discussed using a new representation of the Morse index.