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

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

 (5.18)

where are points inside . In the case of a uniform grid spacing for these points, convergence would be reached for (and resulting error converges like , in numerical experiments), requiring a very large number . In this case all the advantages of a boundary method are lost. However, a Monte Carlo estimate can be reached for a much smaller : in general the estimation error will be Gaussian with a width 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 is such that the time spent evaluating is no more than for , 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 is interesting, since it corresponds to the original recipe of Heller [91]. This recipe involves finding the closest solution to

 (5.19)

where , and at the randomly-chosen interior point . Heller's choice of the , the number of columns of , was about 1.4 times . The density of boundary matching points 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 involving a single interior point. The result is needed (shown in Appendix G). The singular value decomposition of is therefore equivalent to diagonalization of [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 . 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 , 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 stadium billiard at . (Some users have reported as high as 8% in other systems [136]). If a true normalized billiard eigenstate has a very small value of , being times the RMS value, then this state's parabolic tension-minimum curve will be multiplied by (this multiplication is visible in Fig. 5.5). If the dynamic range' (see Section 5.5.3) is less than , 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 occurring are proportional to (the Porter-Thomas probability distribution on is flat around ), one would expect roughly states to be missed. The dynamic range decreases with increasing so it is expected that this will become more severe a problem at high .

Next: Exact form on the Up: The choice of norm Previous: The choice of norm
Alex Barnett 2001-10-03