Brief history of the quantum billiard eigenproblem

A billiard is a closed region of space $\mathcal{D}$ in which the wavenumber $k$ is independent of the position ${\mathbf r} \in \mathcal{D}$. Boundary conditions are defined on $\Gamma$. The number of degrees of freedom (dimensionality of the problem) is $d$. The case of $d=2$ (or occasionally 3) has become a paradigm system for the study of quantum chaos. For this reason, the solution of billiard eigenstates at high energies (quantum number $n \gg 1$) has been of recurring interest in the last 20 years. I will confine myself to study of the case without magnetic field. (I refer the interested reader to the magnetic field billiard method of Smilansky[98]).

Since the billiard interior is a region of free space whose Green's function is known analytically, Class B is the most common approach. Since Class B can reach much higher quantum numbers (being a surface rather than volume method), it is the only viable way to reach the semiclassical regime ( $\lambda_{{\mbox{\tiny B}}}$ of order $L/100$ or less). Two methods have dominated the field. The first is the Boundary Integral Method (BIM), where wavefunctions are represented by a dipole source distribution on the boundary. This was used in the pioneering study by MacDonald and Kaufman[145], using the BIM as presented by Riddell[166]. A similar method was then presented by Berry and Wilkinson [28], which has since been used in an essentially unaltered form [33,137,121]. The BIM still provides the most reliable (it handles a large variety of billiard shapes), if not the most efficient method for billiard eigenproblems. The second is Heller's Plane Wave Decomposition Method (PWDM), which has the edge on efficiency and simplicity over the BIM, mainly due to the simpler basis functions. It has been used apparently in an unaltered form (for instance [136,137,134]) since its invention in 1984 [90,91]. The only significant improvement to this method has been a slightly better search in $E$-space [135].

In 1995 a scaling method which found a large cluster of states in a single diagonalization was invented by Vergini and Saraceno [195]. This is a Class B method which nevertheless returns many states per diagonalization in the manner of Class A methods. It is without doubt the most significant tool in billiard quantization to have emerged in the last 15 years. However its workings are very poorly and briefly explained [195] (much to the lament of many [39]), and there are in fact errors in this explanation. The aim of Chapter 6 is to remedy this situation.

It should be noted that all the above Class B methods have difficulties with certain billiard shapes. The PWDM (and the scaling method) are most sensitive in this respect, because only certain shapes are known for which a good plane-wave basis set exists. Much more study is needed in this area. The BIM suffers from problems with non-convex shapes [137], an active area of research. Currently I believe there is no method reliable for arbitrary shapes, even in $d=2$.

Other, semiclassically-motivated methods have been employed for billiards, for instance KKR [23,162] and the surface of section approaches of Bogomolny [34] and Smilansky [66]. The KKR is applicable only to Sinai-type billiards (cells of periodic lattices). Prosen's matching method [163] (a variant of that of Bogomolny) require the shape to be composed of the union of two integrable regions. As an example of the latter case, the stadium billiard so common in quantum chaos (and in this thesis, see Fig. 2.6) is composed of a circular end-cap joined to a rectangular box, and could be thus solved to arbitrary accuracy without any basis-set choice problems. However, we will see in the next chapter than the efficiency of the scaling method will far exceed any other Class B approach.

Finally, the brute-force Class A approach of Robnik [171,165] should not be overlooked, if only on the grounds of its impressive computational requirements. A banded matrix of order $N=35000$ was diagonalized to give highly accurate low-energy billiard eigenstates (these energies have been a useful standard for error analysis of other methods [137]).