Duality between the trigonometricBC_nSutherland system and a completed rational Ruijsenaars–Schneider–van Diejen system

Abstract

We present a new case of duality between integrable many-body systems, where two systems live on the action-angle phase spaces of each other in such a way that the action variables of each system serve as the particle positions of the other one. Our investigation utilizes an idea that was exploited previously to provide group-theoretic interpretation for several dualities discovered originally by Ruijsenaars. In the grouptheoretic framework one applies Hamiltonian reduction to two Abelian Poisson algebras of invariants on a higher dimensional phase space and identifies their reductions as action and position variables of two integrable systems living on two different models of the single reduced phase space. Taking the cotangent bundle of U(2n) as the upstairs space, we demonstrate how this mechanism leads to a new dual pair involving the BC n trigonometric Sutherland system. Thereby we generalize earlier results pertaining to the A n trigonometric Sutherland system as well as a recent work by Pusztai on the hyperbolic BC n Sutherland system.

The action–angle dual of an integrable Hamiltonian system of Ruijsenaars–Schneider–van Diejen type

“…As was mentioned in the Introduction, the first systematic investigation of action-angle duality relied on direct methods [13,14]. Since then, the reduction interpretation of most (although still not all) examples of Ruijsenaars have been found, and also several new cases of action-angle duality were unearthed utilizing this method; see [16,17,24,25] and references therein. The present paper should be seen as a contribution to the research goal to describe dual pairs for all RSvD type systems in reduction terms.…”

“…It is easy to check that H r has a limit as r → 0. Indeed, we obtain We note for completeness that [25] adopted the inessential condition ν > |κ| ≥ 0.…”

“…lim r→0 H r (λ, θ; u, v, µ) = H 0 (λ, θ; u, v, µ) (A.3) with H 0 (λ, θ; u, v, µ) = V 0 (λ; u, v, µ) +The limiting Hamiltonian H 0 can be recognised as the action-angle dual of the standard trigonometric BC n Hamiltonian. The latter was derived in[25] by reduction of the cotangent bundle of T * U(2n), and was denoted there byH 0 . Concretely, the correspondence with the notations used in equation (1.4) of[25] isH 0 (λ, θ; u, v, 2µ) =H 0 (λ, ϑ; κ, ν, µ) (A.6) under the substitutions u → −κ, v → ν, θ → ϑ.…”

“…The latter was derived in[25] by reduction of the cotangent bundle of T * U(2n), and was denoted there byH 0 . Concretely, the correspondence with the notations used in equation (1.4) of[25] isH 0 (λ, θ; u, v, 2µ) =H 0 (λ, ϑ; κ, ν, µ) (A.6) under the substitutions u → −κ, v → ν, θ → ϑ. (A.7)…”

Self-Duality and Scattering Map for the Hyperbolic van Diejen Systems with Two Coupling Parameters (with an Appendix by S. Ruijsenaars)

“…Nevertheless, to close the gap, the last couple of years have witnessed the emergence of some new ideas in the literature to cope with the intricacies posed by the classical van Diejen models. Indeed, by working out Lax matrices for the most general rational variants of the RSvD family associated with the BC-type root systems, the duality properties of these special non-A-type van Diejen models are also settled completely (see [24,25,26]). Prior to our present work, at the level of the classical hyperbolic systems, non-trivial results could be found only in [27], where the 1-particle BC 1 model is studied by direct techniques.…”