Distribution of particles which produces a desired radiation pattern

摘要

If Aq(β,α,k) is the scattering amplitude, corresponding to a potential qL2(D), where is a bounded domain, and eikα·x is the incident plane wave, then we call the radiation pattern the function A(β)Aq(β,α,k), where the unit vector α, the incident direction, is fixed, and k>0, the wavenumber, is fixed. It is shown that any function f(β)L2(S2), where S2 is the unit sphere in and β is the scattering direction, can be approximated with any desired accuracy by a radiation pattern: f(β)-A(β)L2(S2)<ε, where ε>0 is an arbitrary small fixed number. The potential q, corresponding to A(β), depends on f and ε.