On scattering of the Schrödinger operator with non-local potential

We consider the Schrödinger operator of the form $H=$ $=-d^2/dx^2+V$ acting in $L^2(R)$ where $V=\varepsilon W(x)+\lambda (\cdot ,\varphi _0)\varphi _0$ is non-local potential and $W(x),\, \varphi _0(x)$ are decreasing functions for $|x| \to \infty$. The existence and completeness of the wave operators is proved. We investigate the asymptotic behaviour of solutions of the Lippmann–Schwinger equation and study the scattering amplitude.