The quantum fidelity for the time-periodic singular harmonic oscillator

In this paper we perform an exact study of 'quantum fidelity' (also called Loschmidt echo) for the time-periodic quantum harmonic oscillator of the following Hamiltonian: Hg(t):=(P2/2)+f(t)(Q2/2)+(g2/Q2), when compared with the quantum evolution induced by H0(t) (g=0), in the case where f is a T-periodic function and g a real constant. The reference (initial) state is taken to be an arbitrary 'generalized coherent state' in the sense of Perelomov. We show that, starting with a quadratic decrease in time in the neighborhood of t=0, this quantum fidelity may recur to its initial value 1 at an infinite sequence of times tk. We discuss the result when the classical motion induced by Hamiltonian H0(t) is assumed to be stable versus unstable