Creating desired potentials by embedding small inhomogeneities

TL;DR

This paper demonstrates how to create any desired potential in a Schrödinger-type equation by embedding many small inhomogeneities within a domain, effectively controlling the potential through scatterer distribution.

Contribution

It introduces a method to realize arbitrary potentials by embedding small scatterers with controlled properties, providing a rigorous limit process for the potential approximation.

Findings

Any desired potential can be approximated by embedding small scatterers.

The distribution of scatterers determines the resulting potential.

The method applies in three dimensions and has a one-dimensional analogue.

Abstract

The governing equation is $[\nabla^2+k^2-q(x)]u=0$ in $\R^3$. 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 $B_m:=\{x: |x-x_m|<a\}$, such that $q_m=A_m$ in $B_m$, $q_m=0$ outside $B_m$, $1\leq m \leq M$, $M=M(a)$. It is proved that if the number of small scatterers in any subdomain $\Delta$ is defined as $N(\Delta):=\sum_{x_m\in \Delta}1$ and is given by the formula $N(\Delta)=|V(a)|^{-1}\int_{\Delta}n(x)dx [1+o(1)]$ as $a\to 0$, where $V(a)=4\pi a^3/3$, then the limit of the function $u_{M}(x)$, $\lim_{a\to 0}U_M=u_e(x)$ does exist and solves the equation $[\nabla^2+k^2-q(x)]u=0$ in $\R^3$, where $q(x)=n(x)A(x)$,and $A(x_m)=A_m$. The total number $M$ 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.