About the reconstruction of variable coefficients in a multidimensional wave equation (Q1281240)

An inverse scattering problem for the wave equation \[ \mathcal{ L}(u)=\Delta u - n(x) \frac{\partial^{2} u}{\partial t^{2}} - p(x) \frac{\partial u}{\partial t}=0 \tag{1} \] where \(u=u(x,t)\), \(t\in{\mathbb{R}^{1}}\), \(x\in{\mathbb{R}^{N}}\) \((N\geq 3)\), \(\mathcal L\) is an operator, \(\Delta\) is the Laplacian and \(n(x)\), \(p(x)\) are variable coefficients, is considered. A method of reconstruction of the coefficients of the wave equation based on a longwave asymptotic of the Green's function in the many-dimensional case is proposed. The existence and uniqueness theorem for the Green's function of the Schrödinger operator in the whole space \({\mathbb{R}^{N}} (N\geq 3)\) is proved. The physical sense of the inverse problem is simple: a field with a sufficiently small wave number \(k>0\) is generated in points of some set \(\{y \}\in{\mathbb{R}^{N}}\) \((N\geq 3)\) by a point source. The generated field is measured in points of a set \(\{x\}\in{\mathbb{R}^{N}}\) \((N\geq 3)\). Then the coefficients \(n(x)\), \(p(x)\) of equation (1) can be reconstructed.