The model's partial-wave and phase-shift vocabulary for a declared central potential

(a) Framework data. 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, action constant $\mathrm{\hslash }>0$, and its stationary elastic beam model The declared stationary elastic beam model: incident plane wave, scattered wave, current readings and differential and total cross sections at wave number $k>0$; adopt the declared central-potential reading of The declared central-potential model: state space, assigned energy operator, spherical reading and bound-state vocabulary: a declared central potential of the pair is a real bounded measurable $V(x)=v(\mathrm{\mid }x\mathrm{\mid })$ with $v:(0,\mathrm{\infty })\to \mathbb{R}$ bounded, and it is short-range when it lies in the compact-support class of the scattering model. Units are those of Dimensions, SI unit conventions and quantity types of the framework; the wave number $k$ is read in ${\mathrm{m}}^{-1}$ and the energy $E={\mathrm{\hslash }}^2{k}^2\mathrm{/}(2m)$ in joules.

(b) Channel vocabulary. Let $S^2=\{\omega \in {\mathbb{R}}^3:\mathrm{\mid }\omega \mathrm{\mid }=1\}$ be the sphere of spatial directions, with polar surface measure $\sigma$ and angle coordinates $\mathrm{\Phi }(\theta ,\alpha )=(\sin \theta \cos \alpha ,\sin \theta \sin \alpha ,\cos \theta )$; $d\sigma =\sin \theta d\theta d\alpha$ and $\sigma (S^2)=4\pi$. Use exactly the dimensionless spherical harmonics defined in Associated Legendre function normalization: for integers $l\ge 0$ and $0\le q\le l$, set $P_l(t)=\frac{1}{2^{l}l!}\frac{d^l}{d{t}^l}(t^2-1{)}^l,w_{lq}(t)=(-1{)}^q(1-t^2{)}^{q\mathrm{/}2}\frac{d^q}{d{t}^q}{P}_l(t),b_{lq}={({\int }_{-1}^1{w}_{lq}(t{)}^{2}dt)}^{-1\mathrm{/}2}>0.$ The continuous representatives $Y_{lq}:S^2\to \mathbb{C}$ are given by $Y_{lq}(\mathrm{\Phi }(\theta ,\alpha ))=\frac{b_{lq}}{\sqrt{2\pi }}{w}_{lq}(\cos \theta )e^{iq\alpha }(0<\theta <\pi ,0\le \alpha <2\pi ),$ extended continuously to the poles; for $1\le q\le l$ set $Y_{l,-q}=(-1{)}^q\overline{Y_{lq}}$. The same symbols denote their almost-everywhere classes in $L^2(S^2,\sigma ;\mathbb{C})$. Under the declared Countable Choice assumption, Sphere ONB proves that these classes are orthonormal for $⟨F,G⟩={\int }_{S^2}F\bar{G}d\sigma$ and their closed linear span is all of $L^2(S^2,\sigma ;\mathbb{C})$. For a declared central short-range potential $V$, an angular channel $(l,\bar{m})$, with integers $l\ge 0$ and $-l\le \bar{m}\le l$, means the one-dimensional subspace ${\mathrm{span}}_{\mathbb{C}}\{Y_{l\bar{m}}\}$; the partial wave $l$ means the degree-$l$ sector ${\mathcal{H}}_l:={\mathrm{span}}_{\mathbb{C}}\{Y_{lq}:-l\le q\le l\}$. The angular indices $q,\bar{m}$ are distinct from the positive mass $m$.

The radial solutions are those constructed in Partial-wave cross-section convergence. The reduced equation below is read in physical coordinates after energy division: the reduced potential is $w:=2mv\mathrm{/}{\mathrm{\hslash }}^2$ of clause (c) of The declared short-range scattering model: state space, potential classes, assigned energy observables, asymptotic vocabulary and units and the energy-divided spectral value is $k^2$, both in ${\mathrm{m}}^{-2}$, so that the physical parameters $m,\mathrm{\hslash }$ do not enter it explicitly. The model declares the regular radial solution ${\phi }_l$ of the reduced equation $-{\phi }_l^{\mathrm{\prime }\mathrm{\prime }}+(w(r)+\frac{l(l+1)}{r^2}){\phi }_l=k^2{\phi }_l,{\phi }_l(r)\mathrm{/}{r}^{l+1}⟶1(r\to 0),$ and the outgoing radial solution $f_l$ with $f_l(r)=e^{i(kr-l\pi \mathrm{/}2)}(1+O(r^{-1}))(r\to \mathrm{\infty })$ outside the interaction region. Both are maps $(0,\mathrm{\infty })\to \mathbb{C}$ whose functions and first derivatives are locally absolutely continuous; the reduced equation holds almost everywhere, equivalently in distributions on $(0,\mathrm{\infty })$, as in the cited theorem's bounded measurable potential construction. They are not asserted to be elements of the state space.

(c) Channel factor and phase shift. The model declares the channel factor and the phase shift of the partial wave $l$ to be $S_l(k):=\frac{\overline{W({\phi }_l,f_l)}}{W({\phi }_l,f_l)},S_l(k)=e^{2i{\delta }_l(k)},$ where $W({\phi }_l,f_l)={\phi }_l{f}_l^{\mathrm{\prime }}-{\phi }_l^{\mathrm{\prime }}{f}_l$ is the Wronskian and ${\delta }_l(k)$ is the phase shift, a real number read modulo $\pi$; the model reads $\mathrm{\mid }{S}_l(k)\mathrm{\mid }=1$ and calls the channel elastic. The declared channel cross-section reading is ${\sigma }_l(k):=\frac{4\pi }{k^2}(2l+1){\sin }^2{\delta }_l(k),$ read in ${\mathrm{m}}^2$. The existence of the functions and the identities $\mathrm{\mid }{S}_l\mathrm{\mid }=1$ and ${\sum }_l(2l+1){\sin }^2{\delta }_l<\mathrm{\infty }$ are proved on this page and are not presupposed here.

(d) Status and scope. Clauses (b)-(c) are declarations of the pair: names for the objects constructed by the cited mathematics and the reading of the phase shift modulo $\pi$. The model asserts no identification of the labels $l,\bar{m}$ with a measured angular momentum of a material system, no statement about bound states or their count, and no channel decomposition for non-central potentials. Every numerical statement about ${\delta }_l$ is a separate theorem or example of the pair.

• Modulo $\pi$. The phase shift is read modulo $\pi$, since $S_l=e^{2i{\delta }_l}$ determines ${\delta }_l$ only up to an integer multiple of $\pi$; every display of this page that uses ${\delta }_l$ is insensitive to that ambiguity, as ${\sin }^2$ is.
• Non-central potentials. For a non-central declared potential the pair asserts no diagonal channel factor; the on-shell operator of the following theorem is then a general unitary on $L^2(S^2)$.