Transfer matrix for long-range potentials

TL;DR

This paper generalizes the transfer matrix concept to long-range potentials in quantum scattering, linking it to an effective non-unitary two-level system, and develops an approximation scheme validated by an exactly solvable example.

Contribution

It introduces a transfer matrix framework for long-range potentials, connecting it to an effective quantum system and providing an approximation method for scattering problems.

Findings

Derived properties of the transfer matrix for long-range potentials

Established a connection between transfer matrix and an effective non-unitary quantum system

Validated the approximation scheme with an exactly solvable potential

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.