Write

H = - \partial_{x}^{2} + V (x)

. The real potential and periodic boundary conditions make

H

u_{t} = - i H u

\frac{d}{d t} \int_{- 1}^{1} ∣ u ∣^{2} d x = 2 Re \int_{- 1}^{1} \overline{u}, u_{t} d x = 2 Re (- i \int_{- 1}^{1} \overline{u}, H u d x) = 0,

because the expectation of a self-adjoint operator is real. Equivalently, integrating the kinetic term by parts leaves

\int ∣ u_{x} ∣^{2} d x

and the periodic boundary term cancels. Thus the continuous Schrodinger equation preserves its

L^{2}

b

Solution

The centered spatial second difference has error

O (Δ x^{2})

. The symmetric composition

e^{- i (Δ t /2) T_{h}} e^{- i Δ t V_{h}} e^{- i (Δ t /2) T_{h}}

is Strang splitting, so its local splitting error is

O (Δ t^{3})

and its global time order is two. Because both semidiscrete subproblems are solved exactly, the total global error is

O (Δ t^{2} + Δ x^{2}) .

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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