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.