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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 heavilyscarred 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 randomlychosen 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 2norm 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 itit 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 tensionminimum 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 PorterThomas
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
20011003