Following [195,194], the
singular eigenproblem (5.14) is handled by
restriction of the solution vector
to the non-singular subspace of
.
This means that the numerical null-space of
is ignored.
We have seen above (see Fig. 5.4) that this null-space
corresponds to directions in
which do not contribute to
inside the billiard to more than an accuracy of
.
There is an exception: if
is very close to one of the
, then
restriction of tension to
does not guarantee small values
inside, however the only non-small
which can possibly exist is
the eigenfunction
itself which is already represented.
In conclusion it is fine to ignore this subspace.
Diagonalizing gives
where
is
orthogonal
and
.
This representation of
is now truncated by removing
the eigenvalues smaller than
thus:
, assuming
descending eigenvalue order.
is set to the corresponding first
columns of
.
The transformation
Therefore the recipe is to diagonalize , choose
and construct
and
. Then to construct
, which is diagonalized to give
eigenvalues
and unit-norm eigenvectors
, for
. Finally the eigenvectors are rotated back using the
second relation of
(5.17), to give
.
In order to give the correct normalization
,
corresponding to unit wavefunction normalization in the billiard,
the eigenvectors
should be used.
In particular, the `best' match to the BCs is given by the
largest eigenvalue
.
The corresponding tension of this best
normalized wavefunction is then
.
This recipe was used to generate Fig. 5.2 by sweeping over
.
Note that it would also be possible to select the null-space using
a different criterion. For instance since (5.14) is really
symmetric between and
, the numerical null-space of
could be
used instead. Since both are sensitive to
inside the billiard only,
this would be equally valid. However, it will be convenient to remain with the
above choice for reasons apparent in Section 5.4.