A weakly scattering slab of unit width produces a field satisfying u(x) = e^ ikx +…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Partial differential equations and boundary-value problems · Partial differential equations and boundary-value problems

Conditions of Validity, Ch Integral equations, kernels and the Fredholm alternative for the last time

Problem

A weakly scattering slab of unit width produces a field

u(x)

satisfying

u(x) = e^{ikx} + \mu\int_0^1 \sin(x)\cos(\xi)\,u(\xi)\,\mathrm{d}\xi, \qquad 0 \le x \le 1,

where

k

is the free-space wavenumber and

\mu

measures the scattering strength. ((a)) Classify this equation. ((b)) Predict whether

u

depends continuously on the incident field

e^{ikx}

for a typical value of

\mu

, and say what would have to be special about

\mu

for that to fail.

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