In this chapter I have presented the main components of an improved
`sweep' method (so named because a sweep in is required to locate
eigenstates) for billiard diagonalization.
It falls squarely in Class b) of Table 5.1, although
the zero-determinant condition has been replaced by a maximum-eigenvalue one.
The resulting eigenstates once found can be normalized accurately
using the boundary
formula for
presented in Appendix
H.
The method is a simplified but more rigorous version of Heller's
original PWDM, and reduces to
the key component of Heller's
method in the case of a particular choice of
.
Several problems in Heller's method have been solved.
The choice of basis size in Heller's method requires careful adjustment
(
or so), however the generalized eigenproblem
approach of this chapter truncates
automatically.
The method has been reduced to a simpler algorithm which is entirely
expressible on the boundary, and therefore scales with the area of the boundary.
By reducing to an order-
symmetric eigenproblem,
the choice
can be made without
the disadvantage of slowing down any determinant (SVD) evaluations.
This enables very accurate boundary sampling without slowing down the
matrix operations.
and
are no longer parameters whose choice has a sensitive effect
on the success of the method; they simply need to be sufficiently large.
The accurate norm matrix approaches of Section 5.4
gives clean `switching' between
states corresponding to tension minima--this is an improvement over Heller's
formulation, and would allow such minima to
be distiguished up to a higher
.
The scaling of the numerical effort required per found eigenstate is not
hard to compute.
The basis size is of order the semiclassical basis size (
in
,
in higher
).
The matrices
and
are dense, requiring
effort to diagonalize
(5.14) at a single
.
A few of these are required per state found.
Therefore the effort per state found scales like
.
The accuracy in
found is determined entirely by the basis set
limitations: it appears to be an order of magnitude better than the
`tension rounding error'
.
The method of this chapter
breaks down when the maxima of tension reached between values
becomes about equal to the typical tension
minima
, in which case the fraction of `missing states'
becomes large.
Even long before this is reached, close eigenvalues become indistiguishable
(see Fig. 6.5).
Improved basis sets which can reduce
at a given
require the use of evanescent waves:
the reduction can then be by orders of magnitude in certain billiard shapes
(see Fig. 6.6).
Finally, it is clear that there is much information that is being
wasted by the whole `tension minima hunt' approach described here
and very common elsewhere (including in the vast array of engineering
applications of the BIM).
Fig. 5.6 shows the tension eigenvalues
(inverse generalized of (5.14)).
The square root is plotted so that parabolas become
straight lines, for clarity.
(The finite
actually round the bottoms to give hyperbolae).
It is clear that the eigenstates associated with each minimum
are in fact long-lived as
changes over many level-spacings,
and therefore prediction of a large number of eigenstates from a single
diagonalization is possible.
This is the essence of the method of the next chapter.