Using semiclassical approximations and graph-theoretical concepts, a path representation of the energy-dependent Green function in a one-dimensional multiple-well potential is derived. Applying uniform semiclassical methods, this representation is valid for energies close to the barrier maxima, where tunnelling and above-barrier reflection are of importance. Semiclassical quantization is discussed and a closed path representation of the density of states is derived. Applications to wavepacket dynamics and to the quantization of disordered systems are presented.
Based on Maslov's multidimensional semiclassical theory, a semiclassical Green function in a mixed position-momentum space is derived, which can be used to locally fix the semiclassical divergency in singular regions in configuration or momentum space. It is shown that when a semiclassical Green function is transformed from one representation to another, its form is invariant. The mixed representation includes both coordinate and momentum representations of semiclassical Green functions as two special cases.
The author discusses semiclassical approximations that are adapted to given symmetry classes in quantum mechanics. Arbitrary abelian symmetries and also rotational symmetry are treated. Semiclassical approximations are derived for the projected propagator and energy dependent Green's function associated with a given irreducible representation of the symmetry group. From these they derive trace formulae, analogous to the usual trace formula, that determine the energy levels in a given symmetry class in terms of classical orbits.