Creating desired potentials by embedding small inhomogeneities

Abstract

The governing equation is [∇2+k2−q(x)]u=0 in R3. It is shown that any desired potential q(x), vanishing outside a bounded domain D, bounded in D, Riemann integrable, can be obtained if one embeds into D many small scatterers qm(x), vanishing outside balls Bm≔{x:|x−xm|<a}, such that qm=Am in Bm, qm=0 outside Bm, 1≤m≤M, M=M(a). It is proven that if the number of small scatterers in any subdomain Δ is defined as N(Δ)≔∑xm∊Δ1 and is given by the formula N(Δ)=|V(a)|−1∫Δn(x)dx[1+o(1)] as a→0, where V(a)=4πa3/3, then the limit of the function uM(x), lima→0 uM=ue(x), does exist and solves the equation [∇2+k2−q(x)]u=0 in R3, where q(x)=n(x)A(x), A(xm)=Am, and uM(x) is a solution to the equation [∇2+k2−p(x)]u=0, where p(x)≔pM(x) is some piecewise-constant potential. The total number M of small inhomogeneities is equal to N(D) and is of the order O(a−3) as a→0. A similar result is derived in the one-dimensional case.