On the Equivalence of Different Lax Pairs for the Kac-van Moerbeke Hierarchy

TL;DR

This paper proves algebraically that two different Lax pairs for the Kac-van Moerbeke hierarchy produce the same evolution equations, and introduces new recursions for their computation.

Contribution

It provides a simple algebraic proof of the equivalence of two Lax pairs and derives new recursive formulas for the hierarchy.

Findings

Both Lax pairs generate identical hierarchies.

New recursive formulas for hierarchy computation.

Clarification of the algebraic structure underlying the hierarchy.

Abstract

We give a simple algebraic proof that the two different Lax pairs for the Kac-van Moerbeke hierarchy, constructed from Jacobi respectively super-symmetric Dirac-type difference operators, give rise to the same hierarchy of evolution equations. As a byproduct we obtain some new recursions for computing these equations.