Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 341 / 5 / c

Table of contents
- Solution c

Solution

The periodic centered-difference kinetic matrix

T_{h}

is real symmetric, and the sampled real potential

V_{h}

is a real diagonal matrix. Therefore

- i T_{h}

and

- i V_{h}

are skew-Hermitian, so each matrix exponential in the Strang splitting is a unitary matrix. Their product is unitary as well. Hence

∥ u^{n + 1} ∥_{2}^{2} = ∥ u^{n} ∥_{2}^{2}, \sum_{m} ∣ u_{m}^{n} ∣^{2} = constant .

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