Creating desired potentials by embedding small inhomogeneities

The governing equation is $[{\mathrm{\nabla }}^2+k^2-q(x)]u=0$ in ${\mathbb{R}}^3$. It is shown that any desired potential $q(x)$, vanishing outside a bounded domain $D$, can be obtained if one embeds into D many small scatterers $q_m(x)$, vanishing outside balls $B_m:=\{x:\mathrm{\mid }x-x_m\mathrm{\mid }<a\}$, such that $q_m=A_m$ in $B_m$, $q_m=0$ outside $B_m$, $1\le m\le M$, $M=M(a)$. It is proved that if the number of small scatterers in any subdomain $\mathrm{\Delta }$ is defined as $N(\mathrm{\Delta }):={\sum }_{x_m\in \mathrm{\Delta }}1$ and is given by the formula $N(\mathrm{\Delta })=\mathrm{\mid }V(a){\mathrm{\mid }}^{-1}{\int }_{\mathrm{\Delta }}n(x)dx[1+o(1)]$ as $a\to 0$, where $V(a)=4\pi a^3\mathrm{/}3$, then the limit of the function $u_M(x)$, $\lim_{a\to 0}U_M=u_e(x)$doesexistandsolvestheequation$[\nabla^2+k^2-q(x)]u=0$in$\R^3$,where$q(x)=n(x)A(x)$,and$A(x_m)=A_m$\mathrm{.}Thetotalnumber$M$ of small inhomogeneities is equal to $N(D)$ and is of the order $O(a^{-3})$ as $a\to 0$. A similar result is derived in the one-dimensional case.