Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 2 / 33C / iii

Table of contents
- Solution iii

Solution

The first Born approximation for a one-dimensional reflection coefficient follows by replacing the Jost solution inside its Volterra scattering equation by the free wave

e^{ik x}

:

R (k) = \frac{ϵ}{2 ik} \int_{- \infty}^{\infty} e^{- 2 ik z} q (z) d z + O (ϵ^{2}) .

To check the inverse problem, any shallow discrete state contributes only at

O (ϵ^{2})

pointwise, so to first order the Marchenko input is

F (s) = \frac{1}{2 π} \int_{- \infty}^{\infty} R (k) e^{ik s} d k .

Differentiating and inserting the Born approximation gives

F^{'} (s) = \frac{ϵ}{4 π} \int_{R} \int_{R} e^{ik (s - 2 z)} q (z) d z d k + O (ϵ^{2}) = \frac{ϵ}{4} q (s /2) + O (ϵ^{2}),

where the second line is the Fourier inversion theorem. At first order the quadratic integral term in the Marchenko equation may be dropped, so

K (x, y) = - F (x + y) + O (ϵ^{2}) .

The reconstruction formula therefore returns

u (x) = - 2 \frac{d}{d x} K (x, x) = 4 F^{'} (2 x) + O (ϵ^{2}) = ϵ q (x) + O (ϵ^{2}) .

Thus direct and inverse scattering agree to first order.

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version