  Let $\Delta_{\alpha,Y}$ be the bounded from above self-adjoint realization in
$L^{2}({\mathbb R}^{3})$ of the Laplacian with $n$ point scatterers placed at
$Y=\{y_{1},\dots,y_{n}\}\subset{\mathbb R}^{3}$, the parameters
$(\alpha_{1},\dots\alpha_{n})\equiv\alpha\in {\mathbb R}^{n}$ being related to
the scattering properties of the obstacles. Let $u^{\alpha,Y}_{f_{\epsilon}}$
and $u^{\varnothing}_{f_{\epsilon}}$ denote the solutions of the wave equations
corresponding to $\Delta_{\alpha,Y}$ and to the free Laplacian $\Delta$
respectively, with a source term given by the pulse
$f_{\epsilon}(x)=\sum_{k=1}^{N}f_{k}\,\varphi_{\epsilon}(x-x_{k}) $ supported
in $\epsilon$-neighborhoods of the points in $X_{N}=\{x_{1},\dots, x_{N}\}$,
$X_{N}\cap Y=\varnothing$. We show that, for any fixed
$\lambda>\sup\sigma(\Delta_{\alpha,Y})$, there exits $N_{\circ}\ge 1$ such that
the locations of the points in $Y$ can be determined by the knowledge of the
finite-dimensional scattering data operator $F^{N}_{\lambda}:{\mathbb
R}^{N}\to{\mathbb R}^{N}$, $N\ge N_{\circ}$, $$
(F^{N}_{\lambda}f)_{k}:=\lim_{\epsilon\searrow
0}\int_{0}^{\infty}e^{-\sqrt\lambda\,t}\big(u^{\alpha,Y}_{f_{\epsilon}}(t,x_{k})-u^{\varnothing}_{f_{\epsilon}}(t,x_{k})\big)\,dt\,.
$$ We exploit the factorized form of the resolvent difference
$(-\Delta_{\alpha,Y}+\lambda)^{-1}-(-\Delta+\lambda)^{-1}$ and a variation on
the finite-dimensional factorization in the MUSIC algorithm; multiple
scattering effects are not neglected.
