Even and odd coherent states and excitations of a singular oscillator

We introduce even and odd coherent states. The Green function, the transition amplitudes between the energy levels of a singular nonstationary oscillator in the case of constant frequency in the remote past and future and generating functions for these amplitudes are obtained, by a method similar to the usual coherent-states method. The transition amplitudes are expressed in terms of Jacobi polynomials. Various limit cases are considered of the exact formulas obtained. The results obtained for the one-dimensional problem are generalized to the N-dimensional case, and from there to the case of a charge moving in a singular electromagnetic field. The group-theoretical aspect of the problem is discussed and the groups U (1, 1), and U ( N, 1) are shown to be the dynamical groups of the one-dimensional and N-dimensional cases, respectively.