On the one-dimensional Schrödinger equation with a perturbed non-local steplike potential

We consider a Schrödinger operator of the form $H=-\tfrac{d^2}{dx^2}+V$ acting in $L^2(R)$ where $V=V_0\theta (x)+\varepsilon (\cdot ,\varphi _0) \varphi _0$ is non-local potential. We prove that the unique level (i.e. eigenvalue or resonance of the operator $H$) exists for all sufficiently small $\varepsilon $ and $V_0=V_0(\varepsilon)$. We investigate the asymptotic behaviour of this level. (If $V_0(\varepsilon)$ is separated from zero the levels are absent.) We study the asymptotic behaviour of eigenfunctions for $|x|\to \infty$.