Equivalence of two sets of Hamiltonians associated with the rational BC(n) Ruijsenaars-Schneider-van Diejen system

TL;DR

This paper proves the equivalence between two different sets of commuting Hamiltonians for the rational BC(n) Ruijsenaars-Schneider-van Diejen integrable system, providing an explicit linear transformation between them.

Contribution

It establishes the linear relationship between van Diejen's Hamiltonians and those from the Lax matrix characteristic polynomial, clarifying their equivalence.

Findings

Van Diejen's Hamiltonians are linear combinations of Pusztai's Hamiltonians.

An explicit formula for the linear transformation is derived.

The equivalence enhances understanding of the system's integrability structure.

Abstract

The equivalence of two complete sets of Poisson commuting Hamiltonians of the (super)integrable rational BC(n) Ruijsenaars-Schneider-van Diejen system is established. Specifically, the commuting Hamiltonians constructed by van Diejen are shown to be linear combinations of the Hamiltonians generated by the characteristic polynomial of the Lax matrix obtained recently by Pusztai, and the explicit formula of this invertible linear transformation is found.