Scattering phase shift (${\delta }_l$)

For elastic scattering by a real central potential, the phase shift ${\delta }_l(k)$ measures the phase difference between the asymptotic radial wave in angular-momentum channel $l$ and the corresponding free wave. Its channel Partial-wave S-matrix is $S_l=e^{2i{\delta }_l}$.

The phase shift parametrizes elastic partial-wave scattering, rather than general scattering phenomena.

For regular low-energy elastic scattering by a finite-range central potential, higher partial waves are suppressed by their centrifugal barriers. Matching the regular interior radial solution to small-argument exterior spherical Bessel functions gives the displayed scaling of scattering phase shift, in the generic absence of a threshold singularity. Since the channel amplitude is proportional to $\sin {\delta }_{\ell }/k$, it scales as $k^{2\ell }$. The $\ell =0$ term is isotropic and normally leads. A tuned vanishing leading amplitude or a threshold resonance requires examining the next terms rather than applying the generic statement blindly.

For a reflectionless scattering wavefunction whose two asymptotic phases differ by $\delta (k)$, a large periodic box gives $k_{n}L+\delta (k_n)=2\pi n$. 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.