Next: Equal wavenumber
The following is an application of Green's theorem.
I start with the vector identity
Integrating over , applying Gauss' Theorem to the LHS and Helmholtz
to the first term on the RHS gives
Note that is an abbreviation for
, the local
normal derivative operator at a location on ,
and should not be confused with the cartesian derivatives
, , etc which appear later.
Subtraction of the same equation with the swap
useful overlap formula
This formula is undefined for , where the integral vanishes.
Note that if and are chosen to obey the same (general mixed) BCs,
the formula gives zero as one would expect for orthogonal eigenstates.