  We address the sensitivity of quantum mechanical time evolution by
considering the time decay of the Loschmidt echo (LE) (or fidelity) for local
perturbations of the Hamiltonian. Within a semiclassical approach we derive
analytical expressions for the LE decay for chaotic systems for the whole range
from weak to strong local perturbations and identify different decay regimes
which complement those known for the case of global perturbations. For weak
perturbations a Fermi-golden-rule (FGR) type behavior is recovered. For strong
perturbations the escape-rate regime is reached, where the LE decays
exponentially with a rate independent of the perturbation strength. The
transition between the FGR regime and the escape-rate regime is non-monotonic,
i.e. the rate of the exponential time-decay of the LE oscillates as a function
of the perturbation strength. We further perform extensive quantum mechanical
calculations of the LE based on numerical wave packet evolution which strongly
support our semiclassical theory. Finally, we discuss in some detail possible
experimental realizations for observing the predicted behavior of the LE.
