Uniqueness theorem for inverse scattering problem with non-overdetermined data

Let $q(x)$ be real-valued compactly supported sufficiently smooth function, $q\in H^\ell_0(B_a)$, $B_a:=\{x: |x|\leq a, x\in R^3$ . It is proved that the scattering data $A(-\beta,\beta,k)$ $\forall \beta\in S^2$, $\forall k>0$ determine $q$ uniquely. here $A(\beta,\alpha,k)$ is the scattering amplitude, corresponding to the potential $q$

Creating materials with a desired refraction coefficient

A method is given for creating material with a desired refraction coefficient. The method consists of embedding into a material with known refraction coefficient many small particles of size $a$. The number of particles per unit volume around any point is prescribed, the distance between neighboring particles is $O(a^{\frac{2-\kappa}{3}})$ as $a\to 0$, $0<\kappa<1$ is a fixed parameter. The total number of the embedded particle is $O(a^{\kappa-2})$. The physical properties of the particles are described by the boundary impedance $\zeta_m$ of the $m-th$ particle, $\zeta_m=O(a^{-\kappa})$ as $a\to 0$. The refraction coefficient is the coefficient $n^2(x)$ in the wave equation $[\nabla^2+k^2n^2(x)]u=0$

Uniqueness of the solution to inverse scattering problem with scattering data at a fixed direction of the incident wave

Let $q(x)$ be real-valued compactly supported sufficiently smooth function. It is proved that the scattering data $A(\beta,\alpha_0,k)$ $\forall \beta\in S^2$, $\forall k>0,$ determine $q$ uniquely. Here $\alpha_0\in S^2$ is a fixed direction of the incident plane wave

Creating desired potentials by embedding small inhomogeneities

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

Recovery of a quarkonium system from experimental data

For confining potentials of the form q(r)=r+p(r), where p(r) decays rapidly and is smooth for r>0, it is proved that q(r) can be uniquely recovered from the data {E_j,s_j}, where E_j are the bound states energies and s_j are the values of u'_j(0), and u_j(r) are the normalized eigenfunctions of the problem -u_j" +q(r)u_j=E_ju_j, r>0, u_j(0)=0, ||u_j||=1, where the norm is L^2(0, \infty) norm. An algorithm is given for recovery of p(r) from few experimental data