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).