Estimation by interior points--relation to Heller's PWDM

The simplest approximation to $G$ involves approximating the domain integral by a sum over $M_I$ interior points:

$$G_{ij}(k) \; = \; \frac{{\mathsf{V}}}{M_I} \sum_{m=1}^{M_I} \phi_i(k;{\mathbf r}_m) \phi_j(k;{\mathbf r}_m) ,$$ (5.18)

where ${\mathbf r}_m$ are points inside $\mathcal{D}$. In the case of a uniform grid spacing $a$ for these points, convergence would be reached for $ka \ll 1$ (and resulting error converges like $a$, in numerical experiments), requiring a very large number $M_I \gg {\mathsf{V}}k^d$. In this case all the advantages of a boundary method are lost. However, a Monte Carlo estimate can be reached for a much smaller $M_I$: in general the estimation error will be Gaussian with a width $M_I^{-1/2}$ as a fraction of the value. This requires that the points be chosen in `statistically independent' regions of the wavefunction, a criterion which for ergodic systems (excluding heavily-scarred states) it is easy to fulfill. The slow convergence means that accuracy cannot be high. Generally an optimal choice for $M_I$ is such that the time spent evaluating $G$ is no more than for $F$, preserving the `boundary' scaling of the method. This method has the advantage of simplicity: only basis function values are required (the exact form below requires values and derivatives).

The case $M_I = 1$ is interesting, since it corresponds to the original recipe of Heller [91]. This recipe involves finding the closest solution to

$$\left\{ \begin{array}{c} A(k) {\mathbf x} \; = \; {\mathbf ... ...}^{{\mbox{\tiny T}}} {\mathbf x} \; = \; 1 \end{array} \right.$$ (5.19)

where $A_{mn}(k) = \phi_n(k;{\mathbf s}_m)$, and $y_n = \phi_n(k;{\mathbf r}_1)$ at the randomly-chosen interior point ${\mathbf r}_1$. Heller's choice of the $N$, the number of columns of $A$, was about 1.4 times $N_{sc}$. The density of boundary matching points ${\mathbf s}_m$ was about 2.6 per wavelength. It is easy to see that taking the 2-norm of the above equations gives (5.12) for the version of $G$ involving a single interior point. The result $F = A^{{\mbox{\tiny T}}} A$ is needed (shown in Appendix G). The singular value decomposition of $A$ is therefore equivalent to diagonalization of $F$ [81]. Looking at Fig. 5.5 one can see that the effect of some of the eigenstate dips is almost gone, depending on the choice of ${\mathbf r}_1$. Also notice that the transition from one state to another is not very `clean'. Even though Heller's prescription was then followed by another normalization step and a hunt for the resulting normalized tension minima, if an eigenstate never appears in the solution of the above for any $k$, then this normalization step cannot recover it--it becomes a `missing state'.

What is the expected fraction of such missing states in Heller's method? An exact answer is hard, but empirically seems to be of the order of 1% for the $d=2$ stadium billiard at $k < 400$. (Some users have reported as high as 8% in other systems [136]). If a true normalized billiard eigenstate has a very small value of $\psi({\mathbf r}_1)$, being $\varepsilon$ times the RMS value, then this state's parabolic tension-minimum curve will be multiplied by $1/\varepsilon$ (this multiplication is visible in Fig. 5.5). If the `dynamic range' $r_d$ (see Section 5.5.3) is less than $1/\varepsilon$, then the parabola will typically be `blown' to large enough values that it is obscured by those of neighboring states, and will never be noticed. Since the chances of $\vert\varepsilon \vert < r_d$ occurring are proportional to $1/r_d$ (the Porter-Thomas probability distribution on $\psi{({\mathbf r})}$ is flat around $\psi = 0$), one would expect roughly $1/r_d$ states to be missed. The dynamic range decreases with increasing $k$ so it is expected that this will become more severe a problem at high $k$.