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.