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 tunnelling diode
(RTD) in tilted fields. The RTD is an 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 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 here for the first time
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 successfully treat phenomena beyond
standard periodic orbit (PO) theory: `ghost regions' where no
real PO can be found and regions with contributions from non-isolated
POs. 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.