Periodic-box phase-shift quantization
= Periodic-box phase-shift quantization
{title2=$k_nL+\delta(k_n)=2\pi n$}
For a reflectionless <scattering wavefunction> whose two asymptotic phases differ by $\delta(k)$, a large periodic box gives $k_nL+\delta(k_n)=2\pi n$. Compared with the free value $k_n^{(0)}=2\pi n/L$, $k_n-k_n^{(0)}=-\delta(k_n^{(0)})/L+O(L^{-2})$ away from thresholds. Discrete <bound states> and phase-branch changes alter mode indexing and must be counted when comparing complete spectra.