Source identities and kernel functions for deformed (quantum) Ruijsenaars models

We consider the relativistic generalization of the quantum $A_{N-1}$ Calogero-Sutherland models due to Ruijsenaars, comprising the rational, hyperbolic, trigonometric and elliptic cases. For each of these cases, we find an exact common eigenfunction for a generalization of Ruijsenaars analytic difference operators that gives, as special cases, many different kernel functions; in particular, we find kernel functions for Chalykh- Feigin-Veselov-Sergeev-type deformations of such difference operators which generalize known kernel functions for the Ruijsenaars models. We also discuss possible applications of our results.

💡 Research Summary

The paper investigates the relativistic generalisation of the quantum Aₙ₋₁ Calogero‑Sutherland models introduced by Ruijsenaars, covering the rational, trigonometric, hyperbolic and elliptic regimes. The authors introduce a family of analytic difference operators S⁺ₙ and S⁻ₙ, parametrised by a coupling constant g>0, a relativistic deformation parameter β>0, and a set of auxiliary parameters mₖ drawn from the discrete set Λ={m₀,−m₀,−1/(g)·m₀,1/(g)·m₀}. The function s(x) encodes the underlying potential and takes four different explicit forms depending on the regime (rational, trigonometric, hyperbolic, elliptic). In the elliptic case s(x) is proportional to a Weierstrass sigma function.

The central result is the construction of a common eigenfunction Φ(X;m)=∏_{j<k}φ(X_j−X_k; m_j,m_k) for the whole family of operators. The two‑particle building block φ is expressed in terms of a yet‑to‑be‑specified auxiliary function G(x;α) that satisfies the functional equation

  G(x+iα/2) G(x−iα/2)=c s(x)  (10)

with a non‑zero constant c. For the rational case, G can be chosen as a shifted Gamma function, G(x;α)=Γ(½+x/(iα)), while in the elliptic case G is built from theta functions. The authors show that, once a suitable G is fixed, φ can be written explicitly (formula (9)), covering all four possibilities for the pair (m,m′)∈{(m,m), (m,−m), (m,1/g·m), (m,−1/g·m)}.

Theorem 1.1 states that

  (S⁺ₙ(X;m)−s(i g β∑_j m_j) i g β s′(0)) Φ(X;m)=0

(and the analogous statement for S⁻ₙ). In the elliptic regime this identity holds only if the “balancing condition” ∑_j m_j=0 is satisfied. The proof relies on two auxiliary lemmas. Lemma 2.1 provides a product identity for the s‑function that reduces a product over all pairs to a single factor s(γ∑ m_j) (with γ=i g β), again requiring the balancing condition in the elliptic case. Lemma 2.2 solves a pair of analytic difference equations simultaneously by expressing the solution as a product of two G‑functions, thereby justifying the choice of φ.

With the eigenfunction in hand, the authors derive a plethora of kernel function identities. By specialising the parameters mₖ they recover known kernel functions for the original Ruijsenaars models (including those appearing in the literature on B Cₙ and Toda‑type relativistic systems) and, more importantly, they obtain new kernel functions for Chalykh‑Feigin‑Veselov‑Sergeev (CFVS) type deformations of the Ruijsenaars operators. These deformations involve replacing some of the particles by “dual” variables ˜x and adjusting the coupling constants accordingly; the corresponding kernel functions are given in Corollary 3.7.

Section 4 analyses the non‑relativistic limit (g→0, β→0). In this limit the auxiliary parameters mₖ are no longer confined to Λ and may be arbitrary non‑zero complex numbers, as stated in Theorem 4.1. The authors compare their results with earlier works