General solution of the three-site master equation and the discrete Riccati equation

TL;DR

This paper derives the general solution to a discrete Riccati equation, establishing its equivalence to a second order linear difference equation, with applications to three-site master equations in random walks.

Contribution

It provides a novel general solution to the discrete Riccati equation and links it to second order linear equations, extending supersymmetric quantum mechanics concepts to discrete systems.

Findings

Explicit solutions for free random walk cases

Explicit solutions for biased random walk cases

Establishment of equivalence between Riccati and linear difference equations

Abstract

We first obtain by analogy with the continuous (differential) case the general solution of a discrete Riccati equation. Our results can be considered the discrete analog of Mielnik's construction in supersymmetric quantum mechanics [J. Math. Phys. 25, 3387 (1984)]. Moreover, we establish the full equivalence between our discrete Riccati equation and a corresponding homogeneous second order discrete linear equation. We present an application to the three-site master equation obtaining explicitly the general solutions for the simple cases of free random walk and the biased random walk