Exact treatment of operator difference equations with nonconstant noncommutative coefficients

TL;DR

This paper derives an exact, explicit formula for solving second order difference equations with nonconstant, noncommutative operator coefficients in Hilbert spaces, demonstrating broad applicability through various examples.

Contribution

It provides the first explicit non-iterative solution formula for such operator difference equations with noncommuting coefficients.

Findings

Derived an explicit resolutive formula for the difference equation

Applied the formula to multiple non-trivial examples

Showed broad applicability of the method

Abstract

In this paper we study in a Hilbert space a homogeneous linear second order difference equation with nonconstant and noncommuting operator coefficients. We build its exact resolutive formula consisting in the explicit non-iterative expression of a generic term of the unknown sequence of vectors of the Hilbert space. Some non-trivial applications are reported with the aim of showing the usefulness and the broad applicability of our result.