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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 36B / c / Solution

Here

k cot δ_{0} = - K

, so the s-wave amplitude is

f_{0} (k) = \frac{1}{- K - ik} .

It has a pole at

k = i K

, corresponding to the bound-state energy

E = - \frac{ℏ ^{2} K ^{2}}{2 m} .

Moreover,

σ_{T} ≃ 4 π ∣ f_{0} ∣^{2} = \frac{4 π}{K ^{2} + k ^{2}} .

Solved by gpt-5.6-sol high.

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