Chapters 5 and 6: Improved billiard quantization methods

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 $d=2$ (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 $\hbar\rightarrow0$. 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,

$$(\nabla^2 + k^2) \psi{({\mathbf r})}\ = \ 0,$$ (1.3)

with certain boundary conditions. This problem is common to many other areas of physics and engineering (mesoscopic devices, acoustics, elastodynamics of thin plates, scalar electromagnetics and optics), and there has been some (but not that much) exchange of ideas between those communities and that of quantum chaos. If the boundary conditions are open then we have a scattering problem; if closed, an eigenvalue problem. I will be concerned only with the latter. In Section 5.1 I present a review, and categorize solution methods dependent on whether the basis does not (Class A) or does (Class B) depend on energy. Only Class B allows formulation as a boundary problem. The pioneering chaotic eigenstate studies of McDonald and Kaufman [145] and Berry and Wilkinson [28] used the Boundary Integral Method [36,121] (BIM or BEM), while those of Heller [90,91] used the Plane Wave Decomposition Method (PWDM, an original technique). Semiclassical quantization methods have also been developed based on boundary matching [23] or the surface of section [34]. All these methods are Class B, and all require an expensive search (`sweep' or `hunt') in energy-space for zero-determinants of a matrix.

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 ${\mathbf x}$ of the nearest eigenfunction to a given wavenumber $k$ is given by the largest-eigenvalue ($\lambda_1$) solution to

$$\left[ G(k) - \lambda F(k) \right] {\mathbf x} \ = \ {\mathbf 0} ,$$ (1.4)

where the matrix $G$ takes the norm in the domain, and $F$ takes the norm of the boundary condition error (the `tension'). I show that $G$ can be expressed entirely on the boundary, and discuss improved `hunt' methods for zeros of $\lambda_1^{-1}$ (which give the desired eigenwavenumbers $k$). However, a few diagonalizations of (1.4) are still required per state found.

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-$\lambda$ solutions of

$$\left[ \frac{dF}{dk}(k) - \lambda F(k) \right] {\mathbf x} \ = \ {\mathbf 0} ,$$ (1.5)

however, through a certain choice of boundary weighting function an amazing property of $F$ and $dF/dk$ emerges: they are quasi-diagonal (have very small off-diagonal elements) in a basis of the exact eigenfunctions rescaled to all have the same wavenumber $k$. This allows (1.5) to return up to $N/10$ useful eigenfunctions for a single diagonalization, and entirely eliminates the need for `hunt' procedures. Here $N$ is the matrix size (semiclassical basis size). The relative efficiency over sweep methods is $\sim 10^3$ when there are several hundred wavelengths across the system, and moreover, increases further with increasing $k$ and dimension $d$! Remarkably, no adequate explanation of the key quasi-diagonality property has been known until now. I give, for the first time, a semiclassical explanation in terms of the `special' nature of the dilation deformation (from Chapter 3). I also correct errors in the original authors' derivation [195,194] of higher-order tension terms.

Both chapters are presented as a practical `how-to' guide to the diagonalization of $d$-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.