Scattering problem for the one-dimensional discrete Schrödinger operator with a decreasing potential

We consider the one-dimensional discrete Schrödinger operator $H_0+V$ acting on the space $l^2(\mathbb{Z}),$ where $V$ is a decreasing potential. The theorem of existence and uniqueness of the corresponding Lippmann–Schwinger equation is proved. We study the asymptotics behaviour of solutions of this equation.