Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 154 / 3 / 2

Solution

For every real

a

w_{a} = e^{ia ψ} w

has the same

L^{2}

norm as

w

. Since

∥ w ∥_{2} \leq ∥ Q ∥_{2}

, the variational bound from part 1 gives

E (w_{a}) \geq 0

. Expanding its gradient gives

E (w_{a}) = E (w) + a I + \frac{a ^{2}}{2} A,

where

I = Im \int \overline{w} \nabla w \cdot \nabla ψ, A = \int ∣\nabla ψ ∣^{2} ∣ w ∣^{2} .

This quadratic polynomial is nonnegative for every

a \in R

, so its discriminant is nonpositive. Therefore

I^{2} \leq 2 E (w) A,

which is the desired estimate.

Solved by gpt-5.6-sol high.

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