Joint Eigenfunctions for the Relativistic Calogero–Moser Hamiltonians of Hyperbolic Type II. The Two- and Three-Variable Cases

Abstract

In a previous paper we introduced and developed a recursive construction of joint eigenfunctions J N (a + , a − , b; x, y) for the Hamiltonians of the hyperbolic relativistic Calogero-Moser system with arbitrary particle number N . In this paper we focus on the cases N = 2 and N = 3, and establish a number of conjectured features of the corresponding joint eigenfunctions. More specifically, choosing a + , a − positive, we prove that J 2 (b; x, y) and J 3 (b; x, y) extend to globally meromorphic functions that satisfy various invariance properties as well as a duality relation. We also obtain detailed information on the asymptotic behavior of similarity transformed functions E 2 (b; x, y) and E 3 (b; x, y). In particular, we determine the dominant asymptotics for y 1 − y 2 → ∞ and y 1 − y 2 , y 2 − y 3 → ∞, resp., from which the conjectured factorized scattering can be read off.

Exact quantization conditions for the elliptic Ruijsenaars-Schneider model

“…which again is a complicated infinite-order finite-difference operator on R. However, we already know from previous examples [31,[51][52][53][54][55]70] that when the integrable system is relativistic and has an underlying "modular double" structure, we should in addition consider a "modular dual" Baxter equatioñ and look for two generically linearly independent functions Q (f ) (σ, a, ̵ h y , g y ,ω,ω ′ ) and Q (af ) (σ, a, ̵ h y , g y ,ω,ω ′ ) formally satisfying both (3.72) and (3.73) simultaneously. 18 Similarly to Section 2.5, such formal solutions can conjecturally be obtained by considering the NS limit of the VEVs of codimension two defects corresponding to coupling the fivedimensional theory to two free three-dimensional N = 2 * hypermultiplets, whose SU (2) part of the flavour symmetry is gauged and identified with the five-dimensional gauge group [61][62][63].…”

“…around the hyperbolic limit, since the 2-eCM A and 2-hCM problems have different Hilbert spaces and in addition 2-hCM eigenfunctions are not bound states. Quantum Separation of Variables was developed as a general framework to overcome these and other similar difficulties, and it was successfully applied to the N -particle Toda chain (open and closed) [15][16][17], the N -particle hyperbolic Calogero-Moser system [40], as well as the relativistic version of the open Toda chain [31] and the hyperbolic Ruijsenaars-Schneider system [51,52]. 6 Roughly, the idea of the quantum Separation of Variables approach is to construct the N -particle elliptic Calogero-Moser (of type A) eigenfunction by means of a particular integral transformation which involves the eigenfunction of the N − 1-particle hyperbolic Calogero-Moser system (with center of mass not decoupled) as well as an appropriate entire, fast-decaying solution to an auxiliary 1-dimensional problem known as Baxter equation, which can also be thought of as a quantized version of the spectral curve of the classical integrable system.…”

“…From the works [51,52,65], it appears that the proper way to fix these quasi-constant ambiguities is to study the modular double version of the problem, similarly to what happens for the relativistic open Toda chain [31]; that is, we should require ψ (H) (y) to be a solution not only of the original 2-hRS HamiltonianĤ (H) (3.35), but also of a "modular dual" 2-hRS HamiltonianĤ (H) , related to the original one by the exchange 2ω ←→ ̵ h y : is also related to the original one (3.36) by the exchange 2ω ←→ ̵ h y . SinceĤ (H) is a finite-difference equation in 2ω, requiring (3.37) in addition to (3.35) will completely fix the quasi-constant ambiguity of the solution ψ (H) (y), which should then be 2ω ←→ ̵ h y symmetric [51,52]. We will later see how modular duality plays an important role also for the 2-eRS A system.…”

Wave function for $GL(n,\mathbb{R})$ hyperbolic Sutherland model

“…After this work was completed, we found that M. Hallnäs and S. Ruijsenaars wrote a series of papers [4,5,6], where they suggest a general construction of eigenfunctions for Ruijsenaars systems based on a precise kernel function found in [13]. In particular, degeneration of their construction to hyperbolic Sutherland system [5] yields another presentation of the wave functions given by iterated beta integrals over space variables.…”

Wave function for $GL(n,\mathbb{R})$ hyperbolic Sutherland model II. Dual Hamiltonians

“…On the other hand, shifts in difference operators (1.4) are performed in real direction, which means that the relation (1.5) is understood in a sense of analytical continuation or as an integral equation with Cauchy type kernels. Besides, we have by [KK] the relations Note that M. Hallnäs and S. Ruijesenaars also obtained integral presentations for the wave functions of relativistic Ruijsenaars system [HR2,HR3]. So far we did not establish precise connection of the rational degeneration of these presentations with the formula (1.3).…”