On levels of the one-dimensional Schrödinger operator on the boundary of the essential spectrum

We consider the one-dimensional Schrödinger operator $H$ with the non-local perturbed step potential. We prove that there exists the unique level (i.e. eigenvalue or resonance of the operator $H$) in the neighborhood of the boundary of the essential spectrum of the operator $H$. We investigate the asymptotic behaviour of this level.