Transfer matrix for long-range potentials

Abstract

We extend the notion of the transfer matrix of potential scattering to a large class of long-range potentials v(x) and derive its basic properties. We outline a dynamical formulation of the time-independent scattering theory for this class of potentials where we identify their transfer matrix with the S-matrix of a certain effective non-unitary two-level quantum system. For sufficiently large values of |x|, we express v(x) as the sum of a short-range potential and an exactly solvable long-range potential. Using this result and the composition property of the transfer matrix, we outline an approximation scheme for solving the scattering problem for v(x). To demonstrate the effectiveness of this scheme, we construct an exactly solvable long-range potential and compare the exact values of its reflection and transmission coefficients with those we obtain using our approximation scheme.