The case can be reached most easily by setting
and
while taking the limit
.
The solution now has to be spatially *scaled*
(dilated about the arbitrary origin) appropriately for its new wavenumber.
Section 6.1.1 (also Appendix I)
gives the expansion of the dilation operator
at wavenumber in powers of , thus

This can be shown to be a generalization of the overlap formula given by Boasman [33] to general and differing solutions and . Despite the derivation using scaling, (H.7) is in fact an identity, which can be proved using a messy algebra sequence

In the case of Dirichlet BCs the second term in (H.7) vanishes,
and the replacement
(which follows from the fact that and are parallel),
gives

This very useful boundary formula for the norm of Dirichlet eigenstates appears to have been found first for by Berry and Wilkinson (appendix of [28]). It was since derived in a different way by Boasman[33], and for general was derived in our work[14]. However, I believe the new derivation above to be the simplest yet.