In this appendix we derive the relation (7.23) formally in a self-contained fashion. No reference to the semiclassical derivation of Section 7.1 is made. Therefore this provides an independent derivation of our main result (7.1).
We first present the special case
corresponding to a
QPC system embedded in a hard wall (at ).
off the unbroken wall
is diagonal in the half-plane partial-wave basis
of Section 7.2.
We can expand the `unscattered wave' in Bessel functions [continuing the
expansion of Eq.(7.20)], to give
Now considering a general wall profile with a general
situation becomes more subtle.
An attempt to expand
in Bessel functions as in (L.1) gives additional terms
which are not part of the half-plane basis (7.13), namely
an (s-wave) term and terms with angular dependence
for all .
This would suggest that the basis (7.13) is incomplete.
However, all these new terms can be expressed as sums of the
functions already present, if we are careful
to consider the large radius limit
maximum angular momentum limit
In this limit
, a given basis state will have
-wavevector near the wall, compared to the wall profile width,
so the `softness' of the wall profile
will not be apparent. Rather, the wall will appear as a hard reflector
We write the entire left side wavefunction
We are still left with the issue of finding the incoming
coefficients given an unscattered wave
The problem is subtle, but can be understood
when we consider the order of limits above:
allows the incoming
of a single plane wave
to become a well-defined,
narrowly-peaked delta-like function about the incident angle.
This single plane wave will also cause a
similar delta peak in the opposite direction, which
it turns out is irrelevant because it contributes instead only
to outgoing .
Thus we have the important result that we can ignore the
reflected wave (making its phase shift
in calculating , because this wave can only contribute
to in the interval
We can use stationary-phase integration (method of steepest descents)
 applied to a single plane wave to show this, and
find the delta strength, as follows.
Taking care with the definitions of angle (see Fig. 7.1a) we have