Title:

Correlated Trajectories in Semiclassical Approaches to Quantum Chaos

This thesis is concerned with the application and extension of semiclassical methods,
involving correlated trajectories, that were recently developed to explain the
observed universal statistics of classically chaotic quantum systems. First we consider
systems that depend on an external parameter that does not change the symmetry
of the system. 'Ve study correlations between the spectra at different values
of the param~ter, a scaled distance x apart, via the parametric spectral form factor
K(r, x). Using a semiclassical periodic orbit expansion, we obtain a small r
expansion that agrees with random matrix theory for systems with and without
time reversal symmetry. Then we consider correlations of the Wigner time delay in
open systems. We study a form factor K (r, x, y, M) that depends on the number
of scattering channels M, the nonsymmetry breaking parameter difference x and
also a symmetry breaking parameter y. TheWigner time delay can be expressed
semiclassically in terms of the trapped periodic orbits of the system, and using a
periodic orbit expansion we obtain several terms in the small r expansion of the
form factor that are identical to those calculated from random matrix theory. The
Wigner time delay can also be expressed in terms of scattering trajectories that
enter and leave the system. Starting from this picture, we derive all terms in the
periodic orbit formula and therefore show how the two pictures of the time delay are
related on a semiclassical level. A new type of trajectory correlation is derived which
recreates the terms from the trapped periodic orbits. This involves two trajectories
approaching the same trapped periodic orbit closely  one trajectory approaches the
orbit and follows it for several traversals, while its partner approaches in almost the
same way but follows the periodic orbit an additional number of times.
