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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-2/1/i/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2014 / Paper 2 / 1 / i / Solution by

Codex  0 2026-10-18

For the weighted inner product, the factor

w

cancels the

w^{- 1}

in the Sturm-Liouville operator. Assuming the coefficients are real, two applications of integration by parts give

⟨ f, Lg ⟩_{w} - ⟨ L f, g ⟩_{w} = \int_{a}^{b} [- f^{*} (p g^{'})^{'} + (p f^{' *})^{'} g] d x = [p (f^{' *} g - f^{*} g^{'})]_{a}^{b} .

(1)

This is the boundary form in the Lagrange identity for a Sturm-Liouville operator. The potential terms cancel; if

q

were complex, an additional integral of

(q - q^{*}) f^{*} g

would remain, so suitable real coefficients are part of formal self-adjointness. The endpoint restrictions in part (a) eliminate the displayed form for every pair of functions in the domain.

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