Representation by a Helmholtz basis

I choose a basis set composed entirely of solutions to (5.5) in $\mathcal{D}$. Therefore they are $k$-dependent (equivalently, energy-dependent), and this means we are joining the Class B methods. Since the basis functions do not obey the BCs we can choose the BCs for which the analytic forms of the functions are simplest, namely, free space. The eigenfunctions can be chosen real-valued everywhere, so we can choose entirely real basis functions. Typical basis functions at wavenumber $k$ include: Real Plane Waves (RPWs), Evanescent Plane Waves (EPWs), and angular momentum states (e.g. the singular $Y_0(kr)$) based at various origins (which if they are singular, must be outside $\mathcal{D}$, or placed on $\Gamma$ as in the BIM basis set). Each of these forms is oscillatory with wavenumber $k$. Following Heller's PWDM, the `default' basis will be RPWs uniformly spaced in angle, containing both sin and cos terms, and suitably symmetrized for the billiard shape. The basis size $N$ will be discussed below. Helmholtz basis sets will be further discussed in Sections 5.3.1 and 6.1.3, and examples described in Appendix J.

Substitution of the $k$-dependent basis representation

$$\psi{({\mathbf r})}\; = \; \sum_{n=1}^N x_n \phi_n(k;{\mathbf r}) .$$ (5.11)

into (5.10) and (5.7) gives

$$\left\{ \begin{array}{l} {\mathbf x}^{{\mbox{\tiny T}}}F(k... ...}^{{\mbox{\tiny T}}}G(k) {\mathbf x} = 1, \end{array} \right.$$ (5.12)

where the $k$-dependent positive definite matrices are given by

$$F_{ij}(k) \equiv \oint_\Gamma \!\! w{({\mathbf s})}d{\mathb... ...\! d{\mathbf r} \, \phi_i(k;{\mathbf r})\phi_j(k;{\mathbf r}).$$ (5.13)

The $F$ matrix can be evaluated using methods in Appendix G, and the evaluation of $G$ is discussed in Section 5.4.

The condition (5.12) is equivalent [7] to finding the maximum-eigenvalue solution to the generalized eigenproblem

$$G(k) {\mathbf x} \; = \; \lambda F(k) {\mathbf x}.$$ (5.14)

The maximum eigenvalue is $\lambda_1$, and $\lambda_1^{-1}$ is then simply the tension $f(k)$ corresponding to the `best' solution. The basis coefficients of this best solution are returned in the eigenvector ${\mathbf x}^{(1)}$.

Essentially, the method proposed in this chapter is then searching for local minima in the inverse largest eigenvalue of (5.14), as a function of $k$. An example sweep in $k$ is plotted in Fig. 5.2. However there are implementation issues without which this specification is all but useless. The rest of the chapter is devoted to such issues.

Figure 5.3: Eigenvalues of the `tension matrix' $F$. Left side: sweep over $k$. Right side: slice at a single $k$, with $\lambda_n$ sorted into decreasing order. The billiard is the quarter stadium of Fig. 2.6. The basis set used was a set of 250 real plane waves, symmetrized so as to fall into the odd-odd symmetry class. The semiclassical basis size at this $k$ is $N_{sc} \approx 164$.

Figure 5.4: Eigenvectors ${\mathbf x}^{(n)}$ of the `tension matrix' $F$ at a single $k$: a) $\lambda^F_n \approx 1.3$, in the `semiclassical region', b) $\lambda^F_n \approx 7 \times 10^{-10}$, in the rapid decay towards the null-space region. The corresponding regions are shown in the previous figure as a) and b). The billiard (shown in outline) and basis are the same as the previous figure, except at $k = 50$.