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
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
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