Chapter 5: Improved sweep methods for billiard quantization

In this chapter I describe a numerical method for finding billiard eigenstates which involves `sweeping' the energy to locate the states. It is a generalization of and improvement upon Heller's Plane Wave Decomposition Method (PWDM) [90,91]. The PWDM was invented over 15 years ago and provided a more powerful and rapid way to find eigenstates in certain billiard shapes than had been available beforehand. In this chapter several problems of the PWDM have been solved (missing states, sensitivity to choice of basis set size and number of matching points), it has been simplified, the accuracy is improved, and the speed has been increased considerably. This chapter is also a useful introduction to the following one, where a much more efficient method (which bypasses the `sweep' altogether) is analysed.

- Introduction and history of quantum eigenproblems
- General definitions
- Categories of existing numerical solution techniques
- Brief history of the quantum billiard eigenproblem

- Definition of the billiard problem
- Representation by a Helmholtz basis
- Numerical rank of the basis set
- Truncating the singular generalized eigenproblem
- Choice of for truncation

- The choice of norm matrix
- Estimation by interior points--relation to Heller's PWDM
- Exact form on the boundary and Dirichlet approximation

- The hunt for local tension minima as a function of
- Form of tension minima and resulting accuracy in
- Finding the nearest minimum
- Dynamic range and sweep method breakdown
- A more intelligent hunt?

- Conclusion and discussion

Alex Barnett 2001-10-03