Baxter Operators in Ruijsenaars Hyperbolic System I: Commutativity of Q-Operators

Baxter Q-operator for the hyperbolic Calogero–Moser system *

“…In the N = 2 case, the results in theorems 1.1 and 1.2 (except the difference equation (13) in the latter), as well as the main ideas behind their proofs, can all be extracted from the paper [HR18]. See also the recent preprint by Belousov et al [BDKK23c] for a detailed account of Q-operators for the 2-particle hyperbolic Calogero-Moser system.…”

“…In a remarkable paper, Belousov et al [BDKK23a] recently obtained a Q-operator for the arbitrary-N hyperbolic relativistic Calogero-Moser system; and, in the follow-up paper [BDKK23b], they established an integral equation analogous to (7) for the joint eigenfunctions of the corresponding quantum integrals constructed via an explicit recursive scheme by the author and Ruijsenaars in [HR14].…”

“…Hence it is natural to expect the existence of a dual Q-operator Q, acting in the spectral variable. In the N = 2 case, such a Q-operator Q can be extracted from [HR18] (see equation ( 177)) and is considered in some detail in [BDKK23c]. Moreover, on a formal level, a suitable degeneration of the Q-operator for the hyperbolic relativistic Calogero-Moser system, as obtained in [BDKK23a], yields a natural candidate for Q in the general-N case.…”

“…In the N = 2 case, such a Q-operator Q can be extracted from [HR18] (see equation ( 177)) and is considered in some detail in [BDKK23c]. Moreover, on a formal level, a suitable degeneration of the Q-operator for the hyperbolic relativistic Calogero-Moser system, as obtained in [BDKK23a], yields a natural candidate for Q in the general-N case. We note that it is not entirely clear how best to adapt the methods used here to this dual case, since a direct analogue of the Heckman-Opdam uniqueness result for joint eigenfunctions is lacking.…”