Generalized eigenproblem

The problem is solved by a forming a generalized eigenequation,

$$\left(\frac{dF}{dk} - \lambda_\mu F\right) {\mathbf x}^{(\mu)} \; = \; {\mathbf 0},$$ (6.25)

which finds the eigenstates of the $k$-derivative of tension, treating the tension quadratic form $F$ as a norm which is held constant. The presence of a norm based on the wavefunction at $\Gamma$ means that null-space vectors can be excluded by the method of Section 5.3.2 with truncation $\epsilon$ set at about $10^{-14}$. It is possible to replace this tension norm $F$ by other norms, for instance the exact or Dirichlet $G$ from the previous chapter, however they will not be quasi-diagonal in the scaling eigenfunction basis. This destroys much of the benefit of the quasi-diagonality of $dF/dk$ in this basis. The power of (6.25), as realised by VS, is that both matrices are quasi-diagonal in this basis. Therefore, simultaneous diagonalization (that is, solving the generalized eigenproblem) of $dF/dk$ and $F$ in the computational basis $\phi_i$ returns a very good approximation to the transformation $X$ into the desired eigenfunction basis.

From (6.20) to lowest order the diagonal elements of $\tilde{F}(k)$ are $2\delta_\mu^2$ and those of $d\tilde{F}/dk$ are $4\delta_\mu$. The ratio gives the generalized eigenvalue $\lambda_\mu$. The prediction for the eigenwavenumber is $k_\mu = k - \delta_\mu$, giving to lowest order (see Section 6.2 for higher orders),

$$k_\mu \; = \; k - 2 \lambda_\mu^{-1} + O(\lambda_\mu^{-2}) \cdots .$$ (6.26)

The method will compute all the scaling eigenfunctions within a wavenumber range of up to about 1, for a system size ${\mathsf{L}}\approx 1$. This corresponds to whose rescaled boundaries lie within about $1/12$ wavelength of the original boundary. The predicted $-\delta_\mu$ as a function of $k$ are shown in Fig. 6.4 (bottom).

Figure 6.5: Comparison of eigenvalue solutions returned by a single diagonalization of (6.25) (shown by points, at their resulting $k_\mu$ and tensions $\epsilon_\mu$) against those obtained by the sweep method of the previous chapter (tension shown by line). The agreement is excellent (well within the errors of the sweep minima), apart from the state at $k = 500.0003$ which suffers from error discussed in Section 6.3.2. Close eigenvalues not distiguished by the sweep method are found by the scaling method, even though the same basis set (500 symmetrized RPWs in the stadium) was used. The tensions $t_\mu$ from the scaling method are a factor of 2-3 larger than the sweep method minima $\epsilon_\mu$.

Figure 6.6: Zoom in on Fig. 6.5, with solid line and crosses showing the sweep and scaling methods respectively (basis of 500 RPWs). The dotted line and points show the same with an improved basis (500 RPWs and 30 EPWs). The scaling method returns $k_\mu$ that are well within the `tension rounding error' width $\Delta k(\epsilon_\mu)$ from the sweep method. The evanescent wave improvement is dramatic, and allows the scaling $k_\mu$ to reach accuracy $\sim 10^{-6}$ for the $N/50$ most accurate states.

Figure 6.7: Odd-odd symmetry Dirichlet eigenstate of the 2D stadium billiard at $k=1000.00275996$, shown as a probability density plot. Only the quarter-stadium is shown (spanning about 320 wavelengths). Scarring by the `bow-tie' orbit is visible.