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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 $\gamma$.

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.