Relativistic field-theoretical approach to the inverse scattering problem

The inverse scattering problem for the relativistic three-dimensional equation $\Bigl(2E_{\bf p'}-2E_{\bf p}\Bigr)<{\bf p'}|\Psi_{\bf p}>= -\int V(t)d^3{\bf p''}<{\bf p''}|\Psi_{\bf p}>$ with $E_{\bf p}=\sqrt{m^2+{\bf p}^2}$ and $t=\Bigl(E_{\bf p'}-E_{\bf p''}\Bigr)^2 -\Bigl({\bf p'}-{\bf p''}\Bigr)^2$ is considered. The field-theoretical potential $V(t)$ of this equation is constructed in the framework of the old perturbation theory. It contains all contributions of diagrams with intermediate off-mass shell particles. In particular, this potential reproduces the OBE Bonn model of the $NN$ potential exactly.For the $\pi N$ scattering it is generated by $\sigma,\rho, \omega$-meson exchange diagrams. The inverse scattering problem is solved by reduction of these relativistic equations to the standard Schr\"odinger equations $\Bigl(\Delta_{\bf r}+{\bf k}^2\Bigr)<{\bf r}|\phi_{\bf p}>= - v({\bf r})<{\bf r}|\phi_{\bf k}>$ with $E_{\bf p}={\bf k}^2/2m+m$. The relation between the relativistic potential $V(t)$ and its nonrelativistic representation $v({\bf r})$ is obtained.