  We present an approximation scheme for calculating observables and strength
functions of finite fermionic systems at finite temperature such as hot nuclei.
The approach is formulated within the framework of the Hubbard-Stratonovich
transformation and goes beyond the static path approximation and the RPA by
taking into account small amplitude time-dependent fluctuations around each
static value of the auxiliary fields. We show that this perturbed static path
approach can be used systematically to obtain good approximations for
observable expectation values and for low moments of the strength function. The
approximation for the strength function itself, extracted by an analytic
continuation from the imaginary-time response function, is not always reliable,
and we discuss the origin of the discrepancies and possible improvements. Our
results are tested in a solvable many-body model.
