Limiting absorption principle and the particle current conservation for one-dimensional geometric scattering

Let H₀ be a self-adjoint operator in R^d defined by the Laplacian: H₀=-Δ.. The expression for d>2 the, Green function G₀ (x, y; z) of H₀ (i.e. the integral kernel of the resolvent R₀(z)=(H ₀ -z)^-1) is given. The goal of this paper is to show that at least in the case d⩽3 a relation that expresses the particle current conservation for the one-dimensional geometric scattering in R ^d. Moreover, this relation is valid and has the same physical sense in the case of the Lobachevsky plane or the three-dimensional Lobachevsky space