Spurious surface-wave solutions

Imagine the generalized eigenproblem (6.25) represented in the full linear space of Helmholtz solutions normalizable in $\mathcal{D}$ (as described in Section 6.1.3). This eigenproblem finds the Helmholtz solutions with extremal $df/dk$ under the condition that the tension `norm' $f$ is held constant. The eigenvalue is then $\lambda = (df/dk)/f$. One set of eigensolutions with large $\lambda$ will be the scaling eigenfunctions with $\delta = -2/\lambda$, as demonstrated in Section 6.1.4. However, highly-evanescent surface waves (Section 6.1.3) form another set of eigensolutions, which we call `spurious solutions' (they are not relevant to the Dirichlet billiard problem). We cannot find the exact form of these surface waves, but it is easy to show that they exist and to estimate their $\lambda$, which I now do^6.4.

Take an EPW in this function space, with oscillatory wavenumber $k_{osc}$ and evanescent decay constant $\kappa$. We have $\kappa^2 = k_{osc}^2 - k^2$. If this function is to have a norm of $O(1)$ in $\mathcal{D}$ then it must be oscillating along the boundary, and decaying along the inward normal. (If this were not true, it would have an exponentially-large norm in $\mathcal{D}$. We ignore the complications that may arise in non-convex shapes). It is therefore a surface wave. The contribution to $df/dk$ at a given point on $\Gamma$ is $2\psi \, d\psi/dk$, where $\psi$ is the wavefunction value at that point. The contribution to $f$ is just $\psi^2$. Using $d\psi/dk = (1/k) {\mathbf r} \cdot \nabla \psi$ with $\nabla \psi \approx {\mathbf n} \kappa \psi$ for the EPW, gives $\lambda_{{\mbox{\tiny spurious}}} \approx 2 r_n \kappa/k$ for this wave. Therefore arbitrarily-high $\lambda$ solutions exist with arbitrarily short decay lengths $\kappa^{-1}$. We do not know the exact distribution of $\psi$ on the boundary that forms a solution which is extremal, but it is plausible that they exist. The corresponding wavenumber shift is $\delta_{{\mbox{\tiny spurious}}} \approx -(1/r_n)(k/\kappa)$. For $k_{osc} > 2k$ we can substitute $\kappa \approx k_{osc}$.

The absence of spurious eigenvalues in a $\delta$ range $O(1)$ about zero is clear in the bottom plot of Fig. 6.4. However, spurious solutions (horizontal lines) do exist for $\delta < -0.6$ in this example. Why do the spurious solutions not persist all the way to $\delta = 0$ as the above argument would indicate? The answer is that the basis set (in this case RPWs) cannot represent arbitrarily-high $k_{osc}$ EPWs (the required coefficients diverge exponentially [26,63] so the function rapidly falls into the numerical null-space of $F$ and is truncated away). Therefore a limitation on the maximum $k_{osc}$ representable by the basis (using coefficients $< (\epsilon_{mach})^{-1/2} \sim 10^8$) is what allows a finite window of true scaling eigenfunctions to exist either side of $\delta = 0$. If $\delta_{{\mbox{\tiny spurious}}} \approx -0.6$ with $r_n\approx1$ this corresponds to $k_{osc} \approx 2k$.

The `usable' $N/10$ states returned from the diagonalization fall within a window of about $\vert\delta\vert < 0.2$ for the quarter stadium. To keep spurious solutions below this window would therefore suggest limiting $k_{osc} < 5k$. This is in fact very close to the maximum reliable EPW basis state oscillatory wavenumber which I have found to be about $4k$, independent of $k$.

To conclude, spurious solutions are not a limitation in the basis set. Quite the opposite: they are present in the full normalizable Helmholtz function space, but need to be excluded by an appropriate limitation on the $k_{osc}$ representable by the basis set. Even when they do arise, these spurious solutions are easy to detect and ignore, because their `automatic' normalisation (Section 6.2.2) is much less than one and their tension errors are very large.