Wavefunctions, Green functions and expectation values in terms of spectral determinants

We derive semiclassical approximations for wavefunctions, Green functions and expectation values for classically chaotic quantum systems. Our method consists of applying singular and regular perturbations to quantum Hamiltonians. The wavefunctions, Green functions and expectation values of the unperturbed Hamiltonian are expressed in terms of the spectral determinant of the perturbed Hamiltonian. Semiclassical resummation methods for spectral determinants are applied and yield approximations in terms of a finite number of classical trajectories. The final formulae have a simple form. In contrast to Poincaré surface of section methods, the resummation is done in terms of the periods of the trajectories.