Identification of the model's scattering operator with the on-shell stationary operator, and the central per-channel form

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 compact-support-class potential $V$ supported in $B_R$; by Asymptotic completeness and the unitary scattering operator for real bounded compactly supported potentials the model's wave operators exist and $S={\mathrm{\Omega }}_{+}^{\ast }{\mathrm{\Omega }}_{-}$ is unitary on $H_0$ and commutes with the free flow. Write $\sigma$ for surface measure on $S^2$ and $\mathfrak{h}=L^2(S^2,d\sigma )$, and for every $d\in S^2$ let $f_k(\cdot ,d)$ be the model's stationary amplitude of The model's stationary scattering wave, its outgoing amplitude and the differential and total cross-section readings at incident direction $d$, with the operator ${\mathcal{A}}_k$ on $\mathfrak{h}$ defined by $({\mathcal{A}}_{k}a)(\mathrm{\Omega }):={\int }_{S^2}{f}_k(\mathrm{\Omega },d)a(d)d\sigma (d)\mathrm{.}$ 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) the free flow is diagonalized by the free trace transform of the cited mathematics, a unitary $F_0:H_0\to L^2((0,\mathrm{\infty }),dE;\mathfrak{h})$ with $(F_{0}g)(E,\omega )=E^{1\mathrm{/}4}{2}^{-1\mathrm{/}2}\widehat{g}(\sqrt{E}\omega )$ in the energy-divided variables of the scaling correspondence, where this auxiliary $E=k^2$ has units ${\mathrm{m}}^{-2}$ and the physical energy is $({\mathrm{\hslash }}^2\mathrm{/}2m)E$, and the model's scattering operator is carried into multiplication, $F_{0}S{F}_0^{\ast }=M_{S(E)};$

(b) for almost every $E=k^2>0$ the on-shell operator is $S(E)=I+\frac{ik}{2\pi }{\mathcal{A}}_k,$ a unitary operator on $\mathfrak{h}$, and the model's stationary amplitude $f_k(\mathrm{\Omega },d)$ is the kernel of ${\mathcal{A}}_k$;

(c) for $R\in SO(3)$ let $V_R(x):=V(R^{-1}x)$ and $(U_{R}a)(\omega ):=a(R^{-1}\omega )$; then $S[V_R](E)=U_{R}S[V](E)U_R^{-1};$

(d) if $V$ is in addition central, the fiber decomposes in the spherical harmonic basis, with integer labels $l\ge 0$ and $-l\le \bar{m}\le l$, as $S(E)Y_{l\bar{m}}=S_l(k)Y_{l\bar{m}}$, where $S_l$ and ${\delta }_l$ are the model's channel factor and phase shift of The model's partial-wave and phase-shift vocabulary for a declared central potential, equal to the Wronskian ratio $\overline{W({\phi }_l,f_l)}\mathrm{/}W({\phi }_l,f_l)$, and the cross-section reading of Partial-wave expansion, unit-modulus channel factors and the phase-shift cross-section series is the phase-shift series of that channel decomposition.

The statement identifies operators and operators-valued functions of the declared model; it asserts no measurement of $S$ or $S(E)$, no detector model and no empirical claim.

• What the identification is. The identity $F_{0}S{F}_0^{\ast }=M_{S(E)}$ is an operator identity for the declared model; it is the precise sense in which the pair may speak of "the S-matrix at physical energy $({\mathrm{\hslash }}^2\mathrm{/}2m)E$", and it makes no claim that any apparatus measures $S(E)$.
• Non-central potentials. For a non-central potential no individual spherical-harmonic channel is asserted; only the covariance of (c) holds.