Helsinki University of Technology Somersalo/Bingham Mathematics

Mat-1.198 Scattering Theory

8^{th} set of exercises, 2.4.2003

1. Consider the one-dimensional potential scattering problem: Let q: ℝ → ℝ be a piecewise continuous potential, q(x) = 0 for |x| > M, and u a solution of the equation

u′′+ k^{2}u = qu.

Write

u(x) = u_{inc}(x) + u_{sc}(x),

where the incoming field is a left-moving plane wave sent from +∞,

u_{inc}(x) = e^{−ikx}.

Part of this wave is reflected back, part is transmitted through, so that

u(x) = \{ \begin{matrix}Re^{ikx}+ e^{−ikx}, & x > M \\ Te^{−ikx}, & x < −M. \end{matrix}

(a) Write the appropriate radiation condition in this case.

(b) Derive the Lippmann-Schwinger integral equation.

(c) Find the integral representations of the reflection and transmission coefficients R and T.

(d) What is the Born approximations for R and T?

2. Prove that the scattering is unitary:

|R|^{2}+ |T |^{2} = 1.

(Hint: Consider the Wronskian of the solutions u(x) and \overline{u(x)}.)

3. Prove the inversion formula for Abel’s integral equation: If

h(y) = ∫_{0}^{y} \frac{f(t)}{(y − t)^{α}} dt, 0 < α < 1,

then

f(t) = \frac{sinπα}{π} \frac{d}{dt}∫_{0}^{t} \frac{h(y)}{(t − y)^{1−α}} dy.

(Hint: Use Laplace transform and the convolution theorem.)