Uniform semiclassical methods and their applications

This thesis presents several uniform methods improving the accuracy of semiclassical approximations to quantum mechanics when the WKB approximation breaks down due to nonlinear underlying dynamics. In our alternative approach, the complicated Lagrangian manifold supporting a singular WKB wavefunction is replaced by a series of simpler Lagrangian manifolds, and the singular wavefunction is replaced by a convergent series of well-behaved semiclassical wavefunctions. This method is successfully applied to a model of homoclinic tangle, for which the WKB wavefunction has an infinite number of singularities, and to the problem of wave scattering from a corrugated wall.; A generalized replacement-manifold method is employed to find a uniform wavefunction describing coherent branched flow through a two-dimensional electron gas, a phenomenon recently discovered by direct imaging of the current using scanned probed microscopy. While the formation of branches has been explained by classical arguments, the semiclassical simulations necessary to account for the coherence are difficult due to the proliferation of catastrophes in phase space. The problem is solved here by using replacement manifolds with complex momenta. The method is first explained and tested on a single cusp catastrophe and then on a realistic model of the coherent branched flow.; Another uniform method is used to evaluate the quantum fidelity (Loschmidt echo). This method, based on the classical perturbation approximation and the Initial Value Representation, is numerically tractable and gives remarkably accurate results. Our method explicitly contains the Fermi-Golden-Rule and Lyapunov regimes of fidelity decay as well as the “building blocks” of analytical theories of recent literature, and thus permits a direct test of the approximations made by other authors. The thesis ends with a discussion of what remains from the theory of the decay of survival probability and the parametric dependence of the local density of states in one-dimensional Hamiltonian and disordered systems. We show that a remarkably accurate uniform approximation captures the physics of both perturbative and non-perturbative regimes, though it cannot take into account the strong localization effect.