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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 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 Schrodinger 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