What is the minimum required for a given accuracy in (G.2)?
Writing
as a complex Fourier series on the periodic interval
,
The functions and
are wavefunctions
(Helmholtz equation solutions existing in the plane), measured along
a closed curve, which I first take as smooth and of large curvature radius.
So to a first approximation
and
are bandwidth-limited to
,
where the free-space wavenumber is
.
Therefore
is bandwidth-limited to
.
This would suggest that just over
2 samples per free-space wavelength are needed to achieve
high accuracy, and that convergence is exponential beyond this.
Two facts heavily modify this simple semiclassical picture:
1) there exist evanescent wave components in
eigenstates or basis functions (for instance, evanescent basis functions
can oscillate at up to , see Section 6.3.3),
and
2) the boundary may not be infinitely differentiable
(required for exponential convergence),
rather corners (and `kinks') may introduce discontinuities into the function
which give power-law Fourier tails (and power-law convergence).