Truncating the singular generalized eigenproblem

Following [195,194], the singular eigenproblem (5.14) is handled by restriction of the solution vector ${\mathbf x}$ to the non-singular subspace of $F$. This means that the numerical null-space of $F$ is ignored. We have seen above (see Fig. 5.4) that this null-space corresponds to directions in ${\mathbf x}$ which do not contribute to $\psi$ inside the billiard to more than an accuracy of $\epsilon$. There is an exception: if $k$ is very close to one of the $k_\mu$, then restriction of tension to $\epsilon$ does not guarantee small values inside, however the only non-small $\psi$ which can possibly exist is the eigenfunction $\psi_\mu$ itself which is already represented. In conclusion it is fine to ignore this subspace.

Diagonalizing $F$ gives $F = V \Lambda V^{{\mbox{\tiny T}}}$ where $V$ is orthogonal and $\Lambda = \mbox{diag}(\lambda^F_1 \cdots \lambda^F_N)$. This representation of $F$ is now truncated by removing the eigenvalues smaller than $\epsilon$ thus: $\Lambda_r = \mbox{diag}(\lambda^F_1 \cdots \lambda^F_{N_r})$, assuming descending eigenvalue order. $V_r$ is set to the corresponding first $N_r$ columns of $V$. The transformation

$$G{\mathbf x} = \lambda F{\mathbf x} \hspace{0.2in}\hspace{0... ...n}\hspace{0.2in} G'_r {\mathbf x}'_r = \lambda {\mathbf x}'_r$$ (5.15)

then follows, with the truncated $F$-representation of $G$ defined by

$$G'_r = \Lambda_r^{-1/2} V_r^{{\mbox{\tiny T}}} G V_r \Lambda_r^{-1/2} .$$ (5.16)

The eigenvectors are transformed forwards and backwards by

$${\mathbf x}'_r = \Lambda_r^{1/2} V_r^{{\mbox{\tiny T}}} {\m... ...ce{0.2in} {\mathbf x} = V_r \Lambda_r^{-1/2} {\mathbf x}'_r .$$ (5.17)

There is ambiguity in the latter relation since any vector in the numerical null-space could be added to ${\mathbf x}$, but by design this is not numerically significant when it comes to the resulting wavefunction $\psi{({\mathbf r})}$ in $\mathcal{D}$.

Therefore the recipe is to diagonalize $F$, choose $\epsilon$ and construct $\Lambda_r$ and $V_r$. Then to construct $G'_r$, which is diagonalized to give eigenvalues $\lambda_n$ and unit-norm eigenvectors ${\mathbf x}_r^{\prime (n)}$, for $n = 1 \cdots N_r$. Finally the eigenvectors are rotated back using the second relation of (5.17), to give ${\mathbf x}^{(n)}$. In order to give the correct normalization ${\mathbf x}^{{\mbox{\tiny T}}}G {\mathbf x} = 1$, corresponding to unit wavefunction normalization in the billiard, the eigenvectors ${\mathbf x}^{(n)} / \lambda_n^{1/2}$ should be used. In particular, the `best' match to the BCs is given by the largest eigenvalue $n=1$. The corresponding tension of this best normalized wavefunction is then $\lambda_1^{-1}$. This recipe was used to generate Fig. 5.2 by sweeping over $k$.

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 $F$ and $G$, the numerical null-space of $G$ could be used instead. Since both are sensitive to $\psi$ 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.