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.

b

Solution

Here

Δ x = 1/ M

and periodic indexing is taken modulo

2 M

A = M^{2} - 2 10 ⋮ 01 1 - 2 1 ⋱ 0 01 - 2 ⋱ ⋱ \dots \dots ⋱ ⋱ ⋱ 10 0 ⋱ 1 - 2 1 10 ⋮ 01 - 2 .

Since

V

is real diagonal,

H = A - V

is Hermitian. Therefore

i H

\frac{d}{d t} ∥ u ∥_{2}^{2} = u^{*} (i H) u + u^{*} (- i H) u = 0.

Split the generator as

i A - iV

u^{n + 1} = e^{- ihV /2} e^{ih A} e^{- ihV /2} u^{n} .

The two potential half-steps are componentwise multiplications. The circulant matrix

A

λ_{j} = - 4 M^{2} sin^{2} (\frac{πj}{2 M}), j = 0, \dots, 2 M - 1.

j

e^{ih λ_{j}}

, and an inverse transform. Its cost is

O (M lo g M)

and every factor is unitary, so the implementation preserves the discrete norm exactly up to roundoff.

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