Appendix J: Helmholtz basis functions for two-dimensional billiards

We are interested in a set of basis functions appropriate to represent to sufficient accuracy the eigenstates of a billiard close to a given wavenumber $k$. The basis functions are solutions to the Helmholtz equation (5.5) at wavenumber $k$ (so that the diagonalization problem can be reduced to the boundary). The basis need not obey the boundary conditions (BCs)--if it did, we would already have the desired eigenstates--so we can choose to ignore the BCs and consider the basis as composed of free-space solutions. The exception will be when the billiard shape has symmetries: then the basis should obey certain BCs on the lines (or planes) of symmetry (see below in Section J.4). Since the stadium is the quantum system that I have investigated most, the basis sets presented here are optimized for that shape. The issue of Helmholtz basis sets for more general shapes is open, and is sorely in need of investigation.

Note that because the scaling method of Chapter 6 requires it, I consider only the special case of scaling basis sets,

$$\phi_i(k;{\mathbf r}) \; = \; \bar{\phi}_i(k{\mathbf r}) \hspace{0.5in} \forall i ,$$ (J.1)

where the dimensionless position will be written $k{\mathbf r} \equiv \bar{{\mathbf r}} \equiv (\bar{x},\bar{y})$. This is no great restriction for the plane-wave type basis we present here, however, it disallows basis functions (e.g. singular $Y_0(kr)$) `pinned' to the boundary as is required in the BIM.