Helsinki University of Technology Somersalo/Bingham Mathematics

Mat-1.198 Scattering Theory

7^{th} set of exercises, 26.3.2003

1. Let A(k,x̂,α), k > 0 being the wave number, be the scattering amplitude for potential

scattering,

A(k,x̂,α) = ∫_{D}e^{−ikx̂·y}q(y)u(y,k,α)dy,  (1)

where u(y,k,α) is the solution of the Lippmann-Schwinger equation,

u(x,k,α) = e^{ikα·x}− ∫_{D}Φ(x − y,k)q(y)u(y,k,α)dy.

We can extend the wave number to negative values, too, by substituting −k to the Lippmann-Schwinger equation. Here, we assume that q is real valued and if it depends on k as in the acoustic scattering, it is an even function of k. The corresponding solution satisfies no longer the outgoing radiation condition. However, it allows us to extend A to negative wave numbers, too, by the formula (1) above.

Prove that for this extension, we have

A(−k,x̂,α) = \overline{A(k,x̂,α)}= A(−k,−α,−x̂).

You may assume that n = 3.

2. The Born approximation for the scattering amplitude in material scattering is

A(k,x̂,α) ≈ A_{B}(k,x̂,α) = ∫_{D}e^{−ik(x̂−α)·y} q(y)dy,

i.e., A_{B} is the Fourier transform of q evaluated at

ξ = k(x̂− α) ∈ ℝ^{n}.

Assume that the potential q is independent of the wave number k. (This assumption is valid in quantum mechanical scattering.) Then q itself can be reconstructed approximately by applying the inverse Fourier transform on A. The problem is how to integrate A that is a function defined on ℝ_{+} × S^{n−1} × S^{n−1} .

Prove the following integration identities in the case n = 3:

∫_{ℝ^{3}}f(ξ)dξ = \frac{1}{2} ∫_{−∞}^{∞} ∫_{S^{2}}f(k(x̂− α))|x̂− α|^{2}dαk^{2}dk
= \frac{1}{8π} ∫_{−∞}^{∞} ∫_{S^{2}}∫_{S^{2}}f(k(x̂− α))|x̂− α|^{2}dαdx̂k^{2}dk.

Hence, we obtain an approximation for q as

q(x) ≈ q_{B}(x) = ( \frac{1}{2π}) ^{3} \frac{1}{8π} ∫_{−∞}^{∞} ∫_{S^{2}}∫_{S^{2}}e^{ik(x̂−α)·x}A(k,x̂,α)|x̂− α|^{2}dαdx̂k^{2}dk.

(Hint: Let x̂be a fixed parameter. If ξ = k(x̂− α), show that

k = \frac{|ξ|}{2x̂·ξ̂}, α = x̂− 2(x̂·ξ̂)ξ̂,

where ξ̂ = ξ/|ξ|. These formulas can be used to calculate the Jacobian.)

3. What are the corresponding formulas of the previous problem in ℝ^{2}?