Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 341 / 3 / a

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- Solution a

Solution

The equation can be written

u_{t} = i u_{xx} - iV u .

For the squared L2 norm, periodic integration by parts and the reality of

V

give

\frac{d}{d t} \int_{- 1}^{1} ∣ u ∣^{2} d x = 2 Re \int_{- 1}^{1} (i u_{xx} - iV u) \overline{u} d x = 2 Re [- i \int_{- 1}^{1} ∣ u_{x} ∣^{2} d x - i \int_{- 1}^{1} V ∣ u ∣^{2} d x] = 0.

Thus the Schrodinger equation generates a unitary flow and $∥ u (\cdot, t) ∥_{2}$ is constant.

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