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## Differing wavenumbers

The following is an application of Green's theorem. I start with the vector identity

 (H.3)

Integrating over , applying Gauss' Theorem to the LHS and Helmholtz to the first term on the RHS gives
 (H.4)

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 gives the useful overlap formula
 (H.5)

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.

Alex Barnett 2001-10-03