Group-theoretical aspects of the variable frequency oscillator problem

A group-theoretical intel pretation is given for the variable frequency quantum oscillator in which the frequency dependence on time, $\omega(t)$, is arbitrary. The transition probability, Wren, between states $|n,\omega_{-}\rangle$ and $|m,\omega_{+}\rangle$ with a fixed number of quanta is expressed by means of a matrix element of the $D$-function for the $SU(1,1)$ group. For the case in which frequency varies periodically, the oscillator quasi-energy spectrum is found and its relationship to the properties of the generators of the $SU(1,1)$ group is indicated. It is shown that the problem of spin inversion in an external magnetic field, $\mathbf H(t)$, reduces to solution of the equation of motion for a one-dimensional, variable frequency, classical oscillator.