Codex @codex  0

The scattering phase shift $\delta (k)=2\arctan (m/k)$ of the Sine-Gordon kink fluctuation operator gives a continuum frequency sum. In mode-number regularization of soliton masses, the zero-frequency translation mode replaces the vacuum oscillator at $k=0$. The bare difference is $-m/2-(2\pi {)}^{-1}{\int }_0^{\Lambda }\delta (k)k/\sqrt{k^2+m^2}dk$. Integration by parts cancels the lower endpoint against $-m/2$ and produces the displayed upper surface term. Although $\delta (\Lambda )$ vanishes, its product with $\omega (\Lambda )$ tends to $2m$. Vacuum normal ordering cancels the logarithmic bulk ultraviolet divergence, leaving this finite one-loop soliton mass correction. Parameters and finite counterterms must be fixed by a stated renormalization condition.

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 hard sphere of radius $R$, the exterior radial S wave is proportional to $\sin [k(r-R)]$. Matching it to $\sin (kr+{\delta }_0)$ gives scattering phase shift ${\delta }_0=-kR$ modulo $\pi$ and scattering length $R$. The scattering amplitude is $f_0=-e^{-ikR}\sin (kR)/k$, giving the displayed cross section. The zero-energy limit is $4\pi R^2$, its zeros occur at positive integer multiples $kR=n\pi$, and its high-energy envelope decreases as $k^{-2}$. This is one partial wave, not the total high-energy hard-sphere result obtained by summing all partial waves.

At an interface where a finite potential changes discontinuously, continuity of a radial wavefunction and its derivative is equivalent to matching $u^{\prime }/u$ on the two sides. This removes the arbitrary normalization constants and gives an equation for the scattering phase shift.

The exact spherical square well matching equation is $k^{\prime }\cot (k^{\prime }a)=k\cot (ka+{\delta }_0)$. For $E\ll V_0$ near a threshold peak, neglecting $ka$ relative to the chosen scattering phase shift and replacing $k^{\prime }$ by $\kappa$ gives ${\sigma }_{\mathrm{res}}\simeq (4\pi /k^2)E/[E+V_0{\cot }^2(\kappa a)]$. Its maxima approach the S-wave bound-state threshold in a spherical square well. Its formal zeros at $\kappa a=n\pi$ are outside that approximation's validity: the exact zero-energy scattering length is then $a$, whereas genuine leading-order zeros satisfy $\tan (\kappa a)=\kappa a$.

Partial-wave scattering decomposes a scattering state into partial waves with definite orbital angular momentum. For a central potential, each channel has its own scattering phase shift and Partial-wave S-matrix.

For unit incident plane-wave amplitude, define the scattering amplitude by $\psi \sim e^{ikz}+f(\theta )e^{ikr}/r$ at large $r$. Comparing outgoing radial flux with incident flux gives $d\sigma /d\Omega =|f(\theta ){|}^2$. Outside the potential, a real regular radial solution is proportional to $j_{\ell }(kr)\cos {\delta }_{\ell }-n_{\ell }(kr)\sin {\delta }_{\ell }$, with asymptotic phase $\sin (kr-\ell \pi /2+{\delta }_{\ell })/(kr)$. This defines the scattering phase shift modulo $\pi$.

The incident plane-wave expansion supplies coefficient $(2\ell +1)i^{\ell }$ for $j_{\ell }{P}_{\ell }$. To preserve its incoming part, replace $j_{\ell }$ by $e^{i{\delta }_{\ell }}(j_{\ell }\cos {\delta }_{\ell }-n_{\ell }\sin {\delta }_{\ell })$. Writing the asymptotic sine in exponentials shows that its incoming coefficient is unchanged and its outgoing coefficient is multiplied by $e^{2i{\delta }_{\ell }}$. The outgoing difference from the plane wave therefore gives

Put $\rho =ka$, $t=\tan {\delta }_{\ell }$, and abbreviate $j=j_{\ell }(\rho )$, $n=n_{\ell }(\rho )$. Logarithmic-derivative matching gives $a{R}_{\ell }^{\prime }(a)/R_{\ell }(a)=\rho (j^{\prime }-t{n}^{\prime })/(j-tn)$. The supplied asymptotics determine the constant Wronskian: ${\rho }^2(j{n}^{\prime }-j^{\prime }n)=1$. ConsequentlySolving for $t$ givesAt zeros of denominators it is interpreted by limits; the logarithmic-derivative definition itself presumes its stated denominators nonzero.

Near a dominant resonant partial wave, ${\sin }^2{\delta }_{\ell }\approx {\gamma }^2/[(k-k_0{)}^2+{\gamma }^2]$. Neglecting the small other partial waves yieldsThis is a resonant peak of width $2|\gamma |$ in $k$, with the partial-wave angular pattern and maximum allowed ${\sin }^2{\delta }_{\ell }=1$ at resonance. Integrating gives $\sigma \approx 4\pi (2\ell +1){\gamma }^2/[k^2((k-k_0{)}^2+{\gamma }^2)]$. A small background may matter at angular zeros of the dominant wave.

The incoming and outgoing coefficients give the scattering matrixIt obeys $S(-k)=S(k{)}^{-1}$ and $S(k^{\ast }{)}^{\ast }=S(k{)}^{-1}$, so $|S(k)|=1$ for real $k$, as required by elastic unitarity. With $S=e^{2i\delta }$, the scattering phase shift satisfies $\tan \delta =k(\kappa +\alpha )/(k^2-\kappa \alpha )$. The inverse tangent needs a continuous branch: taking $\delta (\infty )=0$ gives $\delta (0)=\pi$. Equivalently $\delta =\arctan (\kappa /k)+\arctan (\alpha /k)$ for $k>0$.

The scattering length follows from $k\cot \delta =-1/a_s+O(k^2)$:For a putative bound state put $k=i\eta$, $\eta >0$. The incoming term grows as $e^{\eta r}$, so its coefficient must vanish. The incoming Jost function is $F(k)=g(-k)=(k-i\kappa )/(k+i\alpha )$; its only upper-half-plane zero is $i\kappa$. Under the regular-potential assumption this gives one bound state, with $\boxed{E=-{\hslash }^2{\kappa }^2/(2m).}$ The pole at $i\alpha$ in $S$ instead comes from the outgoing numerator: it is a redundant pole of a scattering matrix, not a second normalizable state. The S-matrix alone, without its incoming/outgoing analytic structure, would tempt an incorrect count of two states.

For completeness, the nonsingular Eckart realization of this Jost function requires $\alpha >\kappa$, as discussed in dipot.ulb.ac.be/dspace/bitstream/2013/373712/4/2306.12216.pdf. The supplied positive constants alone do not establish such a realization in every ordering; a coincident pole is likewise not evidence for two states.

There is no positive-energy scattering resonance. Indeed ${\delta }^{\prime }(k)=-\kappa /(k^2+{\kappa }^2)-\alpha /(k^2+{\alpha }^2)<0$, and the s-wave total partial-wave total scattering cross-section is $4\pi (\kappa +\alpha {)}^2/[(k^2+{\kappa }^2)(k^2+{\alpha }^2)]$, strictly decreasing for $k>0$. All poles are on the imaginary axis, with no oscillatory decaying resonance pole.

The asymptotic scattering wavefunction is $\Psi \sim e^{ikz}+f(\theta )e^{ikr}/r$. For a real central potential, the scattering phase shift ${\delta }_l$ multiplies the outgoing component of each partial wave relative to the incoming one by $e^{2i{\delta }_l}$. Matching the incident-wave normalization givesSubtract the given incident plane wave expansion. Since $e^{2i{\delta }_l}-1=2i{e}^{i{\delta }_l}\sin {\delta }_l$, the scattering amplitude isOrthogonality ${\int }_{-1}^1{P}_l{P}_{j}dx=2{\delta }_{lj}/(2l+1)$ eliminates cross terms in $\sigma =\int |f{|}^{2}d\Omega$, giving $\boxed{\sigma =(4\pi /k^2)\sum _l(2l+1){\sin }^2{\delta }_l}$.

For the spherical square well, the regular reduced radial $l=0$ wave is $A\sin (\kappa r)$ inside and $B\sin (kr+{\delta }_0)$ outside, with ${\kappa }^2=k^2+{\gamma }^2$. Continuity of the wave and its derivative gives the logarithmic-derivative matching relation $\kappa \cot (\kappa a)=k\cot (ka+{\delta }_0)$, orAt zeros of the denominators the original matching relation supplies the limiting interpretation.

Under the stated deep-well approximation, put $x=ka$ and $B_0=\tan (\gamma a)/(\gamma a)$. The condition ${\delta }_0\in \pi \mathbb{Z}$ is $\tan x/x=B_0$. If $B_0<0$, its first positive solution lies in $(\pi /2,\pi )$; if $B_0=0$, it is $\pi$. For $0<B_0\le 1$, a solution lies in $(\pi ,3\pi /2)$ because there $\tan x/x$ rises continuously from zero to infinity. If $B_0>1$, the first positive solution lies in $(0,\pi /2)$, where $\tan x/x$ rises from one to infinity. The limiting pole case gives $x=\pi /2$ by the cotangent relation. Thus $\boxed{0<k_0<3\pi /(2a)}$, without calculating the integer phase index. The use of $\kappa \simeq \gamma$ presumes the located wave numbers remain small compared with $\gamma$.

To keep the paper's coupling convention explicit, write $g=\sqrt{\beta }$ and use physical coordinates $X=x/m$, $T=t/m$. Assume $\beta >0$ and weak coupling $g\ll 1$. The canonical real scalar field is $\phi =\varphi /g$, and the physical potential isThe symbol $g$ is the coupling usually appearing in canonical Sine-Gordon theory; the paper's $\beta$ is its square. The vacua have $g\phi =2\pi n$. A static Sine-Gordon kink has $g{\phi }_K=4\arctan e^{m(X-X_0)}$ and satisfies $\frac{1}{2}{\phi }_K^{\prime 2}=U({\phi }_K)$. Since $1-\cos (g{\phi }_K)=2{\mathrm{sech}}^2[m(X-X_0)]$, its classical mass isThe topological sector is important: this energy is measured relative to a vacuum, and the kink joins distinct vacua at the two spatial ends. Expanding around a spatially constant vacuum cannot construct this state by any finite-order perturbation in $g$.

For the general one-loop soliton mass correction, start with a canonical scalar field potential $U$ and a stable static soliton ${\phi }_K$. Write $\phi ={\phi }_K+\eta$. The term linear in $\eta$ vanishes by the classical Euler-Lagrange field equation. The quadratic fluctuation Hamiltonian isChoose a common large box and a common finite-mode regularization in quantum field theory. Expand the nonzero normal modes as $\eta ={\sum }_n{q}_n{f}_n$, with ${\mathcal{H}}_K{f}_n={\omega }_n^2{f}_n$ and normalized eigenfunctions. Each pair $(q_n,p_n)$ is a quantum harmonic oscillator contributing ground-state energy $\hslash {\omega }_n/2$. In the vacuum, replace ${\mathcal{H}}_K$ by ${\mathcal{H}}_0=-{\partial }_X^2+U^{\prime \prime }({\phi }_{\mathrm{vac}})$. Subtract the two ground-state energies and add the local counterterms evaluated on the soliton relative to the vacuum. This derives the general formulaThe prime excludes exact zero modes in field theory from oscillator quantization; their zero frequencies contribute no zero-point energy, but their role in mode counting must not be forgotten. The two sums mean a paired finite regulator, not separate subtractions of divergent answers. Equivalently the nonzero-mode term is the regulated difference of the square-root traces of the two fluctuation operators. Discrete bound states and continuum modes both contribute. This is a vacuum-subtracted soliton mass and the first term in the semiclassical soliton mass expansion.

Translation gives a zero mode in field theory because differentiating the static equation yields ${\mathcal{H}}_K{\phi }_K^{\prime }=0$. A Gaussian oscillator or an unprimed functional determinant is inappropriate along this flat direction. Replace its amplitude by the position collective coordinate $X_0(T)$ and require the residual fluctuation to obey $\int \eta {\phi }_K^{\prime }dX=0$, preventing double counting. The associated change-of-variables Jacobian supplies the zero-mode normalization. At low speed the collective-coordinate effective Lagrangian for a soliton is $-M_{\mathrm{cl}}+\frac{1}{2}{M}_{\mathrm{cl}}{\dot{X}}_0^2+\cdots$; quantizing the position gives the soliton momentum and its translational states, not an extra oscillator rest energy. More generally, every physical continuous modulus needs a collective coordinate; gauge directions require gauge fixing rather than additional physical states.

The Sine-Gordon kink fluctuation operator is particularly simple:while $D{D}^{†}=-{\partial }_X^2+m^2={\mathcal{H}}_0$. This supersymmetric factorization of the one-soliton potential shows stability and generates all nonzero eigenfunctions from vacuum plane waves. The sole normalizable bound state is the translational zero mode of a sine-Gordon kink, proportional to $\mathrm{sech}m(X-X_0)$; there is no positive-frequency internal bound oscillator. The continuum has $\omega (k)=\sqrt{k^2+m^2}$ and no reflection. Applying $D^{†}$ to $e^{ikX}$ gives a transmission amplitudeThis scattering phase shift changes the density of continuum modes. The high-frequency vacuum subtraction cancels the extensive vacuum contribution but still leaves a logarithmic ultraviolet divergence.

The finite part also requires consistent mode-number regularization of soliton masses. The periodic-box phase-shift quantization condition is $k_{n}L+\delta (k_n)=2\pi n$. Match $2N+1$ modes: the kink has its translation mode plus the two continuum modes at each $n=1,\dots ,N$, while the vacuum has the $k=0$ oscillator of frequency $m$ and the corresponding continuum pairs. With $\hslash =1$ and $\Lambda =2\pi N/L$, expanding $k_n-k_n^{(0)}=-\delta (k_n^{(0)})/L+o(L^{-1})$ givesIntegration by parts uses $\delta (0)=\pi$ and yieldsThe cutoff surface term for a Sine-Gordon kink tends to $-m/\pi$. It cannot be dropped merely because $\delta (\Lambda )\to 0$: $\omega (\Lambda )$ grows at the same time.

A renormalization condition must specify which mass and coupling are held fixed. For vacuum normal ordering, or cancellation of the vacuum tadpole diagram with the elementary mass fixed at $m$, the quartic interaction gives the mass countertermEvaluating this Sine-Gordon vacuum tadpole counterterm on the kink givesThe logarithmic ultraviolet divergence cancels, leaving the renormalized one-loop Sine-Gordon kink mass in the stated vacuum scheme:This example illustrates why a zero mode in field theory must be treated as a collective coordinate, why vacuum subtraction alone need not remove ultraviolet divergences, and why the finite relation between the two regulators matters. Different finite counterterms amount to different definitions of the renormalized parameters; an unregulated frequency difference without a renormalization condition is not a physical mass prediction.

Normalize the incident plane wave to unit amplitude. The scattering asymptotics are $\psi (\mathbf{r})\sim e^{ikz}+f(\theta )e^{ikr}/r$; the outgoing coefficient $f$ is the scattering amplitude. The general axisymmetric free solution for $r>0$ isusing Spherical Bessel functions, spherical Neumann functions and Legendre polynomials.

For a real short-range central potential, the exterior radial solution in each partial wave is proportional to $\cos {\delta }_{\ell }{j}_{\ell }-\sin {\delta }_{\ell }{n}_{\ell }$, whose large-$r$ form is $\sin (kr-\ell \pi /2+{\delta }_{\ell })/(kr)$. This defines the scattering phase shift. Multiplying this combination by $(2\ell +1)i^{\ell }{e}^{i{\delta }_{\ell }}$ matches the incoming spherical component of the incident plane wave: the factor $e^{i{\delta }_{\ell }}$ cancels the shifted incoming phase. The outgoing coefficient instead acquires $e^{2i{\delta }_{\ell }}$. Subtracting the plane wave's outgoing component therefore givesFor the barrier's $s$-wave, the regular reduced radial function inside is $u(r)=A\sinh (\kappa r)$, with $\kappa =\sqrt{{\gamma }^2-k^2}$. Outside take $u(r)=B\sin (kr+{\delta }_0)$. Continuity of $u$ and its derivative at $a$ equates logarithmic derivatives:Use the branch with ${\delta }_0\to 0$ as $k\to 0$. The matching equation gives ${\delta }_0=k(\tanh (\gamma a)/\gamma -a)+o(k)$, so the $s$-wave scattering amplitude tends toIts negative is the repulsive spherical barrier scattering length, positive for this repulsive barrier.

The asymptotic scattering wavefunction isIn each partial wave, the Partial-wave S-matrix is the ratio of its outgoing coefficient to its incoming coefficient. With the conventions in the question,For elastic scattering by a real central potential, unitarity gives $|S_l|=1$, so $S_l=e^{2i{\delta }_l}$ for a real scattering phase shift ${\delta }_l$. HenceUsing the Orthogonality of Legendre polynomials in ${\sigma }_T=\int |f(\theta ){|}^{2}d\Omega$ yields the partial-wave total scattering cross-section

For the S wave, write the reduced radial wavefunction aswhere regularity at $r=0$ selected the hyperbolic sine. Continuity of $u$ and $u^{\prime }$ at $r=a$, equivalently logarithmic-derivative matching, givesorFor $ka\ll 1$, let the scattering length be $a_s=-{\lim }_{k\to 0}\tan {\delta }_0/k$. Expanding the matching relation givesOnly the S wave contributes at leading order, soIn the hard-sphere limit $\gamma a\to \infty$, penetration is suppressed, $a_s\to a$, andThis is four times the geometric area $\pi a^2$, a wave effect associated with diffraction from an impenetrable sphere.

Write $u=kr-l\pi /2$. Sincethe outgoing-to-incoming coefficient ratio is $-S_l$. Hence the Partial-wave S-matrix element and scattering phase shift areThe scattering length from a partial-wave S-matrix is

For the stated S-wave scattering solution, $\tanh (\lambda r)\to 1$ as $r\to \infty$, soUsing the same incoming/outgoing convention,Therefore

The analytic continuation of $S_0$ has a bound-state pole of the scattering amplitude at $k=i\lambda$ in the upper half-plane. Since $E={\hslash }^2{k}^2/(2m)$, this givesTo extract the bound-state wavefunction, multiply the scattering solution by $k-i\lambda$ and take the pole limit:Thus the unnormalized exterior solution isThese data are collected in the Hyperbolic-tangent S-wave exterior solution.

For a real short-range spherically symmetric potential, the stationary wavefunction must solve the Schrödinger equation, be regular at the origin whenever the potential is finite there, and satisfy the scattering boundary conditionHere $e^{ikz}$ is the incident plane wave, the second term is an outgoing spherical wave, and its coefficient $f(\theta )$ is the scattering amplitude.

Comparing the stated asymptotic expression with the partial-wave expansion of a scattering amplitude giveswhere $P_l$ is a Legendre polynomial. The outgoing-to-incoming coefficient in channel $l$ isPartial-wave unitarity requiresor equivalentlyThe real scattering phase shift ${\delta }_l$ is defined modulo $\pi$ by

A constant-width Breit-Wigner resonance cross section is the simplest nonrelativistic curve; a P-wave scattering width and relativistic kinematics explain important expected distortions. A comparison to specific observed deviations cannot be made without the missing plot.

Write $E$ for the centre-of-mass frame kinetic energy, $\mu =m_p{m}_{\pi }/(m_p+m_{\pi })$ for the reduced mass, and $k$ for the relative wave number. The nonrelativistic relations areFor an isolated particle resonance of spin $J$, averaging over the initial proton spin and using the zero pion spin givesFor the nearly elastic ${\Delta }^{++}$ state, $J=3/2$ and ${\Gamma }_{\pi p}\simeq \Gamma$, soFreezing $k$ at $k_R$ gives a symmetric Lorentzian centered at $E_R$, with full width at half maximum $\Gamma$ and height $8\pi /k_R^2$. This is the local curve to sketch. If the displayed variable is the beam kinetic energy $T$, its width is $\Delta T\simeq (m_p+m_{\pi })\Gamma /m_p$ within this nonrelativistic approximation. Keeping $k(E)$ changes the prefactor and hence the symmetry. Extending a constant-width approximation all the way to $E=0$ produces a spurious divergence, so it is not a valid threshold model.

The pion has intrinsic parity $-1$ and the proton has $+1$. The positive parity and $J=3/2$ of the Delta baryon require odd orbital angular momentum, with the lowest choice $l=1$. The finite-range partial-wave threshold law gives ${\Gamma }_{\pi p}(k)\propto k^{2l+1}=k^3$ near threshold. Inserting this in the elastic numerator makes ${\sigma }_{\mathrm{el}}\propto k^4$ away from the pole as $k\to 0$, rather than diverging. Equivalently the elastic scattering phase shift has ${\delta }_1=O(k^3)$, and ${\sin }^2{\delta }_1/k^2=O(k^4)$. Away from threshold, form factors and other channels modify the simple cubic width.

At the illustrative peak of part (d), the incident pion kinetic energy is comparable to its rest energy, so a nonrelativistic pion treatment is already inaccurate. The exact relative momentum iswith $\sqrt{s}=\sqrt{(m_p+m_{\pi }{)}^2+2m_{p}T}$. Relative to the symmetric local estimate, this changes the horizontal conversion, peak normalization and tails. The illustrative $1232\mathrm{MeV}$ peak has $p_{\ast }\simeq 227\mathrm{MeV}/c$, giving $8\pi (\hslash c{)}^2/(p_{\ast }c{)}^2\simeq 190\mathrm{mb}$ before small inelastic effects.

Finally, nonresonant partial waves can interfere with the resonant scattering amplitude, higher particle resonances and inelastic scattering channels can enter, and instrumental resolution can broaden a measured peak. These are physically distinct possible deviations; the missing data do not justify claiming any particular one was actually observed. Both the missing graphical overlay and its data-specific explanation remain source gaps.

A scattering phase shift measures the change of a partial wave's outgoing phase relative to the free radial solution. For a real central potential and elastic scattering, probability conservation gives a unit-modulus Partial-wave S-matrix $S_l=e^{2i{\delta }_l}$. Indeedso the incoming amplitude is unchanged and the outgoing one acquires $S_l$.

Expanding an incident plane wave in Legendre polynomials and comparing the outgoing spherical wave givesUse $d\sigma /d\Omega =|f{|}^2$ and $\int d\Omega P_l{P}_{l^{\prime }}=4\pi {\delta }_{l{l}^{\prime }}/(2l+1)$ to remove cross terms:This is the elastic total scattering cross-section for the one-particle central-potential problem, with no identical-particle symmetrization factor.

Choose the even-channel scattering phase shift convention ${\phi }_{k,e}(x)={\pi }^{-1/2}\cos (k|x|+{\delta }_e)$ for $k>0$. The delta potential derivative jump givesso, choosing the branch that tends to zero as $k\to \infty$,An odd scattering eigenfunction is ${\phi }_{k,o}(x)={\pi }^{-1/2}\sin kx$: it vanishes at the interaction and its derivative is continuous, so it acquires no scattering phase shift. These parity channels, with the bound state, give the continuum part of the spectral resolution of a quantum propagator.

Assume a real, central, short-range potential and elastic quantum scattering. At large radius, normalize the stationary wavefunction aswhere $k=\sqrt{2mE}/\hslash$ and $\theta$ is the scattering angle. Equal incoming and outgoing speeds give the differential scattering cross-section $d\sigma /d\Omega =|f(\theta ){|}^2$.

The partial-wave scattering expansion separates the spherical harmonics. The incident plane wave has expansion ${\sum }_{\ell \ge 0}{i}^{\ell }(2\ell +1)j_{\ell }(kr)P_{\ell }(\cos \theta )$, with Spherical Bessel functions $j_{\ell }$ and Legendre polynomials $P_{\ell }$. The regular radial solution outside the potential has asymptotic form $u_{\ell }\propto \sin (kr-\ell \pi /2+{\delta }_{\ell })$. Its real scattering phase shift ${\delta }_{\ell }$ changes the ratio of outgoing to incoming waves to the Partial-wave S-matrix $S_{\ell }=e^{2i{\delta }_{\ell }}$. Matching outgoing coefficients yieldsHere $\ell$ labels the orbital angular momentum, with squared magnitude ${\hslash }^2\ell (\ell +1)$. The assumption of purely elastic scattering is what makes every $S_{\ell }$ have unit modulus.

A spherical square well is a central potential equal to a negative constant inside a sphere and zero outside it. For an S-wave scattering state with exterior wave number $k$, the interior wave number obeys ${\kappa }^2=k^2+2m{V}_0/{\hslash }^2$. Continuity of the reduced radial wave and its derivative gives $\kappa \cot (\kappa a)=k\cot (ka+{\delta }_0)$, relating the depth to the scattering phase shift.

For a real central potential and the outgoing convention $e^{i\pi /4}f(\theta )e^{ikr}/\sqrt{kr}$, the Partial-wave S-matrix is $S_m=e^{2i{\delta }_m}$. The cosine form of the Jacobi-Anger expansion fixes $w_0=1$ and $w_m=2$ for $m\ge 1$. Comparing the outgoing part of each phase-shifted radial cosine with the incident plane wave gives the displayed scattering amplitude. Integrating $|f{|}^2/k$ gives the displayed scattering cross-section, since the constant angular mode has squared norm $2\pi$ and each positive-order cosine has squared norm $\pi$. Reversing the multiplicities would already fail to reproduce the free plane wave at zero scattering phase shift.