Linear function space at fixed wavenumber

There is a linear space of functions which are Helmholtz solutions at wavenumber $k$, which have arbitrary boundary conditions on $\Gamma$, and which are normalizable in $\mathcal{D}$. As we saw in Section 5.3.1, the semiclassical size of this space is $N_{sc} \sim (k{\mathsf{L}})^{d-1}$, corresponding to waves which propagate across the entire billiard, for instance real plane waves (RPWs)--plane waves with real wavevector components. One possible represention of this space is by an angular distribution function $f(\Omega)$, namely

$$\psi(k;{\mathbf r}) \; = \; \int \! d\Omega \, \psi(\Omega) e^{ik{\mathbf n}_\Omega \cdot {\mathbf r}} ,$$ (6.16)

where $\Omega$ is the angle coordinate in $d$ dimensions and ${\mathbf n}_\Omega$ the corresponding unit normal vector. Propagating waves correspond to functions $\psi(\Omega)$ which oscillate on an $\Omega$-scale $\sim (k{\mathsf{L}})^{-1}$ but no faster. Another possible representation is using surface `charge' (Green's function monopole or directed-dipole sources) distributed across $\Gamma$--this is the basis set of the Boundary Integral Method (BIM) [36,121]. Again, the spatial frequency of this surface charge is limited (to $k$) for propagating waves.

However there are also infinitely many evanescent waves which cling to the surface without penetrating far into the volume ^6.2 [194]. The higher the imaginary part of the wavenumber (the more evanescent), the quicker is the decay into the interior. Evanescent plane waves (EPWs) fall into this category. EPWs can be represented by distributions of RPWs in the finite region $\mathcal{D}$, however the required coefficients grow exponentially with the evanescence parameter, and are of rapidly alternating sign [26]. This means that evanescent (surface) waves can be represented by angular functions $\psi(\Omega)$ which oscillate more rapidly than $\sim (k{\mathsf{L}})^{-1}$ and are exponentially large. (The corresponding surface charge representation oscillates more rapidly than $k$, however it has the advantage that the charge is not exponentially large).

Therefore if all such evanescent waves are included, the dimensionality of the linear space seems infinite. However this issue is a tricky one because of the exponential divergences, and has only been explored quite recently (see also [64,63]). Practically, the inclusion of some EPWs into a representation of the function space enables the coefficients to avoid exponential blow-up in their closest representation of eigenfunctions (see Appendix J). Therefore in practice, the dimensionality required is not much larger (a factor $1.1$ to $1.2$ larger in the 2D stadium) than $N_{sc}$.

The use of EPWs is very beneficial in many systems, particularly the stadium, where the tension minima $\epsilon_\mu$ can be lowered from the range $10^{-5}$ to $10^{-7}$ to the range $10^{-10}$ to $10^{-12}$ (see Fig. 6.6), a dramatic discovery made by Vergini[194]. Therefore for all high-accuracy computations for the stadium a basis composed of RPWs plus a few EPWs is used. The particular EPWs used [194] are described in Appendix J. It seems that most eigenfunctions of nonintegrable shapes and even some integrable ones have evanescent character [63,194]. The problem of finding suitable evanescent waves to add to the basis for a general billiard shape is unsolved. There are many shapes for which no good basis is known (Sinai billiard, generalised Sinai billiard of Fig. 3.2a, etc.).