Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 32C / d

Table of contents
- Solution d

Solution

∣ x ∣ \to \infty

, the operator

A

tends to

- 4 \partial_{x}^{3}

. Evolve the normalized scattering eigenfunction by

\partial_{t} ϕ_{k} = A ϕ_{k} + 4 i k^{3} ϕ_{k} .

The added scalar term keeps the coefficient of

e^{- ik x}

at

- \infty

equal to one. At

+ \infty

,

(- 4 \partial_{x}^{3} + 4 i k^{3}) e^{- ik x} = 0, (- 4 \partial_{x}^{3} + 4 i k^{3}) e^{ik x} = 8 i k^{3} e^{ik x} .

Consequently

\overset{a}{˙} (k, t) = 0, \dot{b} (k, t) = 8 i k^{3} b (k, t) .

It follows that

T (k, t) = T (k, 0), R (k, t) = R (k, 0) e^{8 i k^{3} t} .

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