The rapid development of electronic computing machines in the last 50 years
has had an impact on scientific research whose size is hard to
grasp^{1.2}.
The interplay between numerical simulations and theoretical models now plays a
crucial role in most areas of physics, chemistry, engineering, and other
sciences.
However, this impact would have been be drastically reduced were it not for
the parallel development of efficient numerical algorithms.
For instance, the invention of two techniques alone--the
diagonalization of dense matrices[81],
and the Fast Fourier Transform[161]--has allowed scientists
to handle hitherto undreamed-of problems on a daily basis.

Quantum chaos [91] is no exception: it has relied heavily
on numerical solutions almost since its inception
(as did its forebear, classical chaotic dynamics[154]).
Billiards in (or sometimes 3) dimensions have been popular systems for
study because
use of the free-space Green's function allows formulation as a boundary problem.
Thus
quantum eigenstates can be calculated at much higher energy than with the
traditional (*e.g. *finite element) methods which cover the entire domain.
High energies are so sought-after because most of the theoretical predictions
involve the semiclassical limit
.
In this part of the thesis I will present new and efficient
methods for finding these high-energy eigenstates.

The time-independent Schrodinger's equation in such a system is
the Helmholtz wave equation,

In Chapter 5 I present an original
Class B sweep method which is a simplified
version of Heller's PWDM. The problems of missing states and sensitivity to
basis size choice and matching point density have been solved,
and the efficieny increased.
The coefficient vector of the nearest eigenfunction to a given
wavenumber is given by the largest-eigenvalue () solution to

Of much more significance is Chapter 6.
Here I analyse the
`scaling method' of Vergini and Saraceno
[195,194],
which despite being little-understood and little-used, is without doubt the
most significant advance in numerical billiard quantization in the last 15 years.
Eigenstates are given by the large- solutions of

Both chapters are presented as a practical `how-to' guide to the diagonalization of -dimensional billiards, and I hope they may be of use to other communities who solve the Helmholtz eigenproblem. I thoroughly analyse the various types of error in both the sweep and scaling methods, compare results from the two, and discuss the use of real and evanescent plane wave basis sets. For illustration, I use Bunimovich's stadium billiard (a shape known to be classically-chaotic [40]), in which evanescent basis sets have been pioneered by Vergini[194]. Currently the scaling method applies only to Dirichlet boundary conditions. Adequate basis sets for more general shapes is an area in dire need of future research. My work has involved deriving a collection of useful new formulae for boundary evaluation of domain integrals of Helmholtz solutions: these are presented in Appendix H.

Two applications of the scaling method are presented in this thesis. The first is the quantum band-profile calculations for Chapters 2-4. The second is an efficient evaluation of overlaps of eigenstates of a billiard with eigenstates of the same billiard deformed by various finite amounts (Section 6.4). The profiles of the resulting matrices can be viewed as local densities of states (`line shapes'), which are analysed in our publication [48]. The diagonalization of the deformed stadium billiard is believed to be new.

During this work, I have benefitted much from fruitful exchanges with Eduardo Vergini. I must thank Doron Cohen for first alerting me to the semiclassical estimation of the band profile of matrix elements on the boundary. Finally, Appendix H resulted from collaboration with Michael Haggerty.