Partial-wave expansion, unit-modulus channel factors and the phase-shift cross-section series

Adopt the declared short-range scattering model The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units with mass $m>0$, spin zero and a declared central short-range potential $V(x)=v(\mathrm{\mid }x\mathrm{\mid })$ in the sense of The model's partial-wave and phase-shift vocabulary for a declared central potential clause (a): $v$ is real, bounded and $V$ is supported in a ball $B_R$. Adopt the stationary elastic beam model at wave number $k>0$ and direction $d\in S^2$, with amplitude $f_k$ as in The model's stationary scattering wave, its outgoing amplitude and the differential and total cross-section readings, and the channel vocabulary of The model's partial-wave and phase-shift vocabulary for a declared central potential clauses (b), (c). Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Then:

(a) $f_k$ depends only on $\cos \theta =\mathrm{\Omega }\cdot d$, and its Legendre expansion $f_k(\theta )=\frac{1}{2ik}{\sum }_{l\ge 0}(2l+1)(S_l(k)-1)P_l(\cos \theta )$ converges in $L^2(S^2)$ and absolutely, so the model's cross-section reading is $\frac{d\sigma }{d\mathrm{\Omega }}(\theta )=\mathrm{\mid }{f}_k(\theta ){\mathrm{\mid }}^2;$

(b) the channel factors are unit modulus, $\mathrm{\mid }{S}_l(k)\mathrm{\mid }=1$, the phase shifts ${\delta }_l(k)$ are real modulo $\pi$, and the total cross section of the beam model is $\sigma =\frac{\pi }{k^2}{\sum }_{l\ge 0}(2l+1)\mathrm{\mid }{S}_l(k)-1{\mathrm{\mid }}^2=\frac{4\pi }{k^2}{\sum }_{l\ge 0}(2l+1){\sin }^2{\delta }_l(k)\text{in }{\mathrm{m}}^2,$ the series converging, with the channel readings of The model's partial-wave and phase-shift vocabulary for a declared central potential clause (c) as its summands;

(c) the objects ${\phi }_l$, $f_l$, $S_l$ and ${\delta }_l$ of the channel vocabulary are exactly the regular and outgoing radial solutions, the Wronskian ratio and the phase shift constructed by the cited mathematics for the reduced radial potential $w=(2m\mathrm{/}{\mathrm{\hslash }}^2)v$ at the model's wave number $k$; the phase shifts are not asserted to have a definite sign and no low-energy limit is asserted beyond the convergence of the series.

The statement is conditional on the declared central model, the beam model and the cited mathematics; it asserts no measured angular distribution and no identification of $l$ with a measured angular momentum.

• Resonance models. A single-channel model of the phase shift is declared and analysed separately on this page (The declared single-channel resonance model: pole-shaped channel factor and its cross-section reading); this theorem asserts nothing about the existence of resonances for a given potential.
• Low-energy behaviour. The series converges for every $k>0$ of the declared class; no statement is made about its behaviour as $k\to 0$, where the class hypotheses themselves remain in force but the pair asserts no limiting formula.