It is possible to calculate *exactly* using basis function
(or derivative) evaluations.
The equivalent expressions (H.14) and (H.7)
both give the domain overlap of two Helmholtz solutions
(with no particular BCs) at the same wavenumber,
purely in terms of boundary integrals.
Taking for example the former expression, it can be approximated
(using the discretization of Appendix G) as

This formidable expression has been coded and tested. It has been found
that for the Dirichlet eigenproblem, it can be replaced by a much
simpler form which is valid only when the wavefunctions in question
vanish on the boundary, namely (H.9).
When written in a basis this gives

where . The largest few resulting generalized eigenvalues (and eigenfunctions) of (5.14) are indistiguishable from those resulting from (5.20). That this works is not entirely obvious, because the basis functions do not vanish on the boundary. However all the relevant vectors contain combinations of basis functions which

As a bonus, the eigenstates with Neumann BCs are to be found at the
minima of the *smallest* .
This is true because exchanging Dirichlet for Neumann in the tension
function definition is equivalent to swapping and in
(5.14).