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Generalized eigenproblem
The problem is solved by a forming a generalized eigenequation,

(6.25) 
which finds the eigenstates of the derivative of tension, treating the
tension quadratic form as a norm which is held constant.
The presence of a norm based on the wavefunction at
means that nullspace vectors can be excluded by the method
of Section 5.3.2 with truncation set at about .
It is possible to replace this tension norm by other norms, for instance
the exact or Dirichlet from the previous chapter, however they will not
be quasidiagonal in the scaling eigenfunction basis. This destroys much of the
benefit of the quasidiagonality of in this basis.
The power of (6.25), as realised by VS,
is that both matrices are quasidiagonal in this
basis.
Therefore, simultaneous diagonalization (that is, solving the generalized
eigenproblem) of and in the computational basis
returns a very good approximation to the transformation into the
desired eigenfunction basis.
From (6.20) to lowest order the
diagonal elements of are and
those of are .
The ratio gives the generalized eigenvalue .
The prediction for the eigenwavenumber is
, giving to lowest order (see Section 6.2
for higher orders),

(6.26) 
The method will compute all the scaling eigenfunctions
within a wavenumber range of up to about 1, for a system size
.
This corresponds to whose rescaled boundaries lie within
about wavelength of the original boundary.
The predicted as a function of 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 and tensions
) 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 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 from the scaling method are a factor of 23 larger than the
sweep method minima .

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 that are well within the `tension rounding
error' width
from the sweep method.
The evanescent wave improvement is dramatic, and allows the scaling
to reach accuracy for the most accurate states.

Figure 6.7:
Oddodd symmetry Dirichlet eigenstate of the 2D stadium billiard
at
, shown as a probability density plot.
Only the quarterstadium is shown (spanning about 320 wavelengths).
Scarring by the `bowtie' orbit is visible.

Next: The scaling method in
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Alex Barnett
20011003