= Solution
For small fixed positive $E$, the stated approximate curve has maxima at
$$
\boxed{\kappa a=\left(n+\frac12\right)\pi,\qquad
\sigma_{\rm res,max}=\frac{2\pi\hbar^2}{mE}=\frac{4\pi}{k^2},
\quad n=0,1,\ldots.}
$$
Its formal minima are zero at $\kappa a=n\pi$. The peak height is the $S$-wave <unitarity> limit, and the peaks narrow as $E/V_0$ decreases.
\Image[/past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2012/paper-1-square-well-resonances.png]
{title=Square-well resonance limits}
To identify the bound-state threshold, let the bound-state <energy> be $-\varepsilon$ with $\varepsilon>0$. The reduced radial wave is proportional to $\sin(Kr)$ inside and $e^{-\chi r}$ outside, with $K=\sqrt{2m(V_0-\varepsilon)}/\hbar$ and $\chi=\sqrt{2m\varepsilon}/\hbar$. Matching gives $K\cot(Ka)=-\chi$. As a new <bound state> emerges at $\varepsilon\downarrow0$,
$$
\boxed{\kappa\cot(\kappa a)=0
\ \Longrightarrow\
\kappa a=\left(n+\frac12\right)\pi.}
$$
Thus the limiting peak locations coincide with the appearance of successive \b[$S$-wave <bound states>]. At the exact threshold the zero-energy radial solution is not <square-integrable>; a <normalizable> <bound state> appears just beyond it. Finite-energy maxima shift slightly from the zero-energy <positions>.
The formal minima require care because the assumption used in part d fails there. Retaining $ka$ in the exact matching condition gives the <spherical square-well scattering length>
$$
a_s=a-\frac{\tan(\kappa a)}{\kappa},\qquad
\sigma_0\longrightarrow4\pi a_s^2
$$
away from a threshold. At $\kappa a=n\pi$, $a_s=a$, so the true low-energy <S-wave scattering> contribution is $4\pi a^2$, not zero. Genuine leading-order zeros of that contribution instead satisfy $\tan q=q$, for example $q\simeq4.493409$ and $7.725252$. The original sketch compares the exact finite-energy $S$ wave with the requested approximation, making this nonuniformity visible. Dominance by the $S$ wave must be checked separately near a higher-partial-wave threshold. The scattering-length divergence and its bound-state interpretation are also derived in institutional lecture notes.<Topics in Quantum Mechanics by David Tong>[2017, §6.2.6, printed pp. 219–222: square-well matching, scattering length and threshold bound states]{ref}
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