Given the isotropic quadratic dispersion relation corresponding
to (5.2), we can choose energy units such that
.
If the dispersion relation is not isotropic, it can be made so by
a re-parametrization of .
However, note that will not play any further role.
The numerical methods described in this thesis are
really about finding the *eigenwavenumbers* .
Therefore the methods are entirely applicable to any other Helmholtz
eigenproblem regardless of the dispersion relation, or indeed the existence
of an `energy' (for instance is physically irrelevant in acoustic problems).
The only requirement is that the wavenumber be constant
(and isotropic) in the interior.

The billiard has -dimensional `volume' and dimensional
`surface area' , giving a typical length scale
.
Our eigenproblem can be written

where the eigenvalue is . The functionals and return a scalar, and are

The BCs have been incorporated as (5.6)
rather than the linear condition (5.3)
because satisfaction of the BCs by a wavefunction is then
revealed by a single number .
This number measures the 2-norm of some error function,
and is therefore a non-negative quantity.
The error function (*e.g. *(5.3)) gives the amount
by which the desired BCs fail to be obeyed.
Heller[91] named `tension', and I shall follow suit.
The definition of is

Without a further condition, (5.5) and (5.6) admit the
useless solution,
for all .
Therefore the quadratic functional

The solution
is now completely determined, when reaches one
of the eigenwavenumbers .
For other , no solution exists.
Therefore in order to be able to define a `best' solution for *any*
given guess
at , one of the conditions needs to be relaxed.
The condition (5.6) will be replaced by the minimization