Factorization method on time scales

TL;DR

This paper introduces a factorization method for second order difference equations on time scales, constructing Hilbert spaces and ladder operators to generate solutions, bridging continuous and discrete analysis.

Contribution

It develops a novel factorization approach for second order difference equations on time scales, including the construction of Hilbert spaces and ladder operators.

Findings

Constructed Hilbert spaces on time scales

Developed ladder operators for difference equations

Generated new solutions via eigenproblem factorization

Abstract

We present an approach to the factorization method for second order difference equations on time scales. We construct Hilbert spaces of functions on the time scale and show how to construct a chain of intertwined first order $\Delta$-difference operators $\mathcal A_k$ in the sense that $\mathcal A_k$ and $\mathcal A_k^*$ act as ladder operators generating new solutions of eigenproblem of $\mathcal A_k\mathcal A_k^*$.