  We derive semiclassical expressions for spectra, weighted by matrix elements
of a Gaussian observable, relevant to a range of molecular and mesoscopic
systems. We apply the formalism to the particular example of the resonant
tunneling diode (RTD) in tilted fields. The RTD is a new experimental
realization of a mesoscopic system exhibiting a transition to chaos. It has
generated much interest and several different semiclassical theories for the
RTD have been proposed recently.Our formalism clarifies the relationship
between the different approaches and to previous work on semiclassical theories
of matrix elements. We introduce three possible levels of approximation in the
application of the stationary phase approximation, depending on typical length
scales of oscillations of the semiclassical Green's function, relative to the
degree of localization of the observable. Different types of trajectories
(periodic, normal, closed and saddle orbits) are shown to arise from such
considerations. We propose a new type of trajectory (``minimal orbits'')and
show they provide the best real approximation to the complex saddle points of
the stationary phase approximation. We test the semiclassical formulae on
quantum calculations and experimental data. We show that the new types of
trajectories (saddle and minimal orbits) provide accurate results. We discuss a
divergence of the contribution of saddle orbits, which suggests the existence
of bifurcation-type phenomena affecting the complex and non-periodic saddle
orbits.
