 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2017/s18/1/6/solution

For

E = - ℏ^{2} κ^{2} / (2 m)

, the decaying Green's function is

m e^{- κ ∣ x - x^{'} ∣} / (κ ℏ^{2})

, with the same derivative jump as before. A normalizable negative-energy state has no incident homogeneous part. Applying the inverse to

L ψ = - U ψ

gives

ψ (x) = - \frac{m}{κ ℏ ^{2}} \int e^{- κ ∣ x - x^{'} ∣} U (x^{'}) ψ (x^{'}) d x^{'} .

The condition

κ > 0

is necessary for decay on both sides.

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