  In this paper we develop the topics of Quantum Recurrences and of Quantum
Fidelity which have attracted great interest in recent years. The return
probability is given by the square modulus of the overlap between a given
initial wavepacket and the corresponding evolved one; quantum recurrences in
time can be observed if this overlap is unity. We provide some conditions under
which this is semiclassically achieved taking as initial wavepacket a coherent
state located on a closed orbit of the corresponding classical motion. The
"quantum fidelity" (or Loschmidt Echo) is the square modulus of the overlap of
an evoloved quantum state with the same evoloved by a slightly perturbed
Hamiltonian. Its decrease in time measures the sensitivity of Quantum Evolution
with respect to small perturbations. It is believed to have significantly
different behavior in time when the underlying classical motion is chaotic or
regular. Starting with suitable initial quantum states, we develop a
semiclassical estimate of this quantum fidelity in the Linear Response
framework (appropriate for the small perturbation regime), assuming some
ergodicity conditions on the corresponding classical motion.
