Hyperbolic Ruijsenaars Model

• Hyperbolic Ruijsenaars model is a relativistic integrable system defined by commuting finite-difference Hamiltonians with hyperbolic coefficients.
• It employs modular-double and Macdonald operator formulations that facilitate explicit spectral analysis and Baxter Q-operator construction.
• The model connects diverse frameworks such as Liouville CFT, quantum Teichmüller theory, and gauge theory through duality and reflection symmetry.

The hyperbolic Ruijsenaars model, usually called the hyperbolic Ruijsenaars–Schneider system, is a relativistic integrable deformation of Calogero–Sutherland-type many-body systems in which the quantum Hamiltonians are commuting finite-difference operators with hyperbolic coefficients. In the $A$-type setting it admits an $N$-particle modular-double formulation with periods $\omega_1,\omega_2$, a distinguished two-particle reduction with explicitly known eigenfunctions, Baxter $Q$-operators, bispectrality, and exact integral transforms; it also appears in Liouville conformal field theory, quantum Teichmüller theory, Poisson–Lie reduction, and gauge-theoretic Coulomb-branch constructions (Belousov et al., 17 Aug 2025, Belousov et al., 2024, Apresyan et al., 25 Mar 2026).

1. Definition and operator realizations

A standard $N$-particle hyperbolic Ruijsenaars Hamiltonian acts on a wave function $\psi(\{x_j\})$ by

$$\bigl(H^{(\omega_1)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_1,\ldots),$$

and it has a commuting partner obtained by interchanging $\omega_1$ and $\omega_2$: $$\bigl(H^{(\omega_2)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_2,\ldots).$$ The two operators commute, and their coefficients are governed by the modular-double periods $N$0 and the coupling $N$1 (Belousov et al., 17 Aug 2025).

In the Macdonald gauge, the same system is described by the commuting family

$N$2

where $N$3 and $N$4. These operators are gauge-equivalent to the Ruijsenaars operators $N$5 via a diagonal factor built from the double sine $N$6: $N$7 The standard parameter domain used in the analytic theory is

$N$8

with the stronger condition $N$9 often imposed for adjointness and contour-separation arguments (Belousov et al., 2023).

The special-function backbone is the hyperbolic gamma or double-sine function. In $\omega_1,\omega_2$0-notation, $\omega_1,\omega_2$1 satisfies

$\omega_1,\omega_2$2

together with the reflection relation

$\omega_1,\omega_2$3

In Liouville notation one writes $\omega_1,\omega_2$4 and $\omega_1,\omega_2$5 (Belousov et al., 17 Aug 2025).

2. The two-particle hyperbolic model

In the two-particle center-of-mass reduction, the hyperbolic Ruijsenaars operator takes a particularly explicit finite-difference form. In the $\omega_1,\omega_2$6-parametrization used in the Liouville-theoretic treatment,

$\omega_1,\omega_2$7

so the shift step is $\omega_1,\omega_2$8. Its distinguished eigenfunction is

$\omega_1,\omega_2$9

and it satisfies the spectral equation

$Q$0

Here $Q$1, $Q$2, the integration variable lies on $Q$3, and the spectrum is continuous with eigenvalues $Q$4 (Apresyan et al., 25 Mar 2026).

The same two-particle system can be written in modular-double variables as

$Q$5

together with the commuting partner

$Q$6

Their joint eigenfunctions are analytic in $Q$7 and can be expressed in Mellin–Barnes form through $Q$8 (Belousov et al., 17 Aug 2025).

A characteristic structural property is self-duality: $Q$9 This exchanges the coordinate and spectral variables while reflecting the coupling $N$0 (Apresyan et al., 25 Mar 2026).

3. Baxter operators, product formulas, and spectral transform

For the two-particle model, the product formula

$N$1

leads directly to an integral Baxter operator $N$2 with kernel built from the same $N$3-factors. Its action on the eigenfunction is diagonal: $N$4 and the operators satisfy

$N$5

In that two-particle Liouville-derived framework, a standard scalar $N$6–$N$7 functional relation is not written explicitly; the characterization is by commutation and diagonal action on $N$8 (Apresyan et al., 25 Mar 2026).

For the $N$9 hyperbolic system, Baxter operators form a commuting family of integral operators

$\psi(\{x_j\})$0

and they commute with the Macdonald operators. The Hallnäs–Ruijsenaars joint eigenfunction $\psi(\{x_j\})$1, constructed recursively by the raising operators $\psi(\{x_j\})$2, diagonalizes them: $\psi(\{x_j\})$3 while simultaneously satisfying the Macdonald spectral problem

$\psi(\{x_j\})$4

The same wave function is therefore jointly diagonal for the commuting hyperbolic Macdonald and Baxter families (Belousov et al., 2023).

Orthogonality and completeness were then established by exploiting Baxter diagonalization and coordinate–spectral duality. The resulting transform is an $\psi(\{x_j\})$5-particle analogue of Fourier inversion: for symmetric functions,

$\psi(\{x_j\})$6

with

$\psi(\{x_j\})$7

In distributional form, the wave functions satisfy orthogonality and completeness relations

$\psi(\{x_j\})$8

and the transform is unitary in four regimes: I: $\psi(\{x_j\})$9, $$\bigl(H^{(\omega_1)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_1,\ldots),$$0; II: $$\bigl(H^{(\omega_1)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_1,\ldots),$$1, $$\bigl(H^{(\omega_1)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_1,\ldots),$$2; III: $$\bigl(H^{(\omega_1)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_1,\ldots),$$3, $$\bigl(H^{(\omega_1)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_1,\ldots),$$4; IV: $$\bigl(H^{(\omega_1)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_1,\ldots),$$5, $$\bigl(H^{(\omega_1)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_1,\ldots),$$6 (Belousov et al., 2023, Belousov et al., 2024).

4. Liouville CFT, quantum Teichmüller theory, and coupling reflection

A central recent result is that the two-particle hyperbolic Ruijsenaars Hamiltonian, its Baxter operator, and the product formula for its eigenfunctions can be derived from the genus-one Moore–Seiberg identity in Liouville conformal field theory. The Liouville parameters are identified by

$$\bigl(H^{(\omega_1)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_1,\ldots),$$7

and the Ruijsenaars wavefunction coincides, up to known normalization factors, with the Liouville modular one-point $$\bigl(H^{(\omega_1)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_1,\ldots),$$8-kernel. A degenerate insertion $$\bigl(H^{(\omega_1)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_2}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_1,\ldots),$$9 reduces the Moore–Seiberg identity to a two-term finite-difference equation reproducing the Ruijsenaars Hamiltonian, while the condition

$\omega_1$0

leads to the exact product formula. The authors further expect extension to $\omega_1$1-particle hyperbolic Ruijsenaars systems via conformal Toda field theory and to supersymmetric variants via super Liouville theory (Apresyan et al., 25 Mar 2026).

A different geometric realization appears in $\omega_1$2 quantum Teichmüller theory on the punctured torus. There, the order-four modular element

$\omega_1$3

acts on the Hilbert space of the quantized cluster Poisson variety, and the matrix coefficient

$\omega_1$4

is a joint eigenfunction of the two-particle hyperbolic Macdonald operators. In this framework the Hallnäs–Ruijsenaars eigenfunction is proportional to the modular $\omega_1$5-matrix coefficient, and $\omega_1$6 Macdonald polynomials arise as special values of its analytic continuation. The construction uses an $\omega_1$7-equivariant embedding of the $\omega_1$8 spherical DAHA into the quantized coordinate ring of the cluster Poisson variety (Francesco et al., 2024).

Coupling reflection is encoded by a second Baxter family. Writing

$\omega_1$9

one introduces $\omega_2$0, conjugate to the original $\omega_2$1 at reflected coupling. The two Baxter families commute with each other and with the Macdonald operators, and the wave function obeys the exact symmetry

$\omega_2$2

This refines the coordinate–spectral duality by adding a direct $\omega_2$3 reflection symmetry at the level of wave functions (Belousov et al., 2023).

5. Classical structure, Lax formulations, and spin generalizations

The classical hyperbolic Ruijsenaars model arises by Poisson reduction from the Heisenberg double of $\omega_2$4. In reduced variables $\omega_2$5, $\omega_2$6, the Lax matrix can be written as

$\omega_2$7

Its spectral invariants

$\omega_2$8

commute, and the Poisson bracket of $\omega_2$9 has quadratic dynamical $$\bigl(H^{(\omega_2)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_2,\ldots).$$0-matrix form. Upon introduction of a spectral parameter, the corresponding baxterized $$\bigl(H^{(\omega_2)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_2,\ldots).$$1-matrices satisfy shifted Yang–Baxter-type equations; on the quantum side, trace-type formulae for commuting integrals are conjectured to generate the same commutative ring as the Macdonald operators (Arutyunov et al., 2019).

A structurally related family is the hyperbolic spin Ruijsenaars–Schneider model. In one Poisson-reduction derivation, the spin degrees of freedom are rectangular matrices

$$\bigl(H^{(\omega_2)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_2,\ldots).$$2

the phase space is built from the Heisenberg double together with a deformed oscillator manifold, and the reduced Lax matrix has collective spin entries $$\bigl(H^{(\omega_2)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_2,\ldots).$$3. The commuting Hamiltonians remain

$$\bigl(H^{(\omega_2)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_2,\ldots).$$4

while the model carries Poisson–Lie symmetry of the spin group $$\bigl(H^{(\omega_2)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_2,\ldots).$$5, providing superintegrability (Arutyunov et al., 2019).

A gauge-theoretic realization identifies the hyperbolic spin model with the $$\bigl(H^{(\omega_2)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_2,\ldots).$$6-theoretic Coulomb branch of 3d $$\bigl(H^{(\omega_2)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_2,\ldots).$$7 affine $$\bigl(H^{(\omega_2)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_2,\ldots).$$8 necklace quiver gauge theory. In that description the positions are $$\bigl(H^{(\omega_2)}_{\mathrm{RS}}\psi\bigr)(\{x_j\}) =\sum_{j=1}^{N}\left[\prod_{k\neq j}\frac{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k+i g)\Bigr)}{\sinh\!\Bigl(\tfrac{\pi}{\omega_1}(x_j-x_k)\Bigr)}\right]\psi(\ldots,x_j+i\omega_2,\ldots).$$9, the relativistic coupling is $N$00, the Hamiltonians are

$N$01

and the equations of motion reproduce the Krichever–Zabrodin hyperbolic spin RS system with potential

$N$02

The same construction exhibits the classical limit of the quantum toroidal algebra of $N$03, and the Hamiltonians lie in the center of this symmetry, making superintegrability manifest (Arutyunov et al., 3 Mar 2026).

6. Related families, lattice versions, and degenerations

The hyperbolic Ruijsenaars model sits inside a larger family of Ruijsenaars–Schneider–van Diejen systems. In the two-parameter hyperbolic van Diejen model,

$N$04

the interaction factor $N$05 contains both $N$06 and $N$07 terms as well as a one-body contribution depending on $N$08. Setting $N$09 removes the one-body term, and suppressing the $N$10-type sum interactions recovers the $N$11-type hyperbolic RS Hamiltonian structure. The same work establishes global action–angle variables, self-duality, and a factorized scattering map for the van Diejen system, thereby extending Ruijsenaars-type phenomena beyond the translation-invariant $N$12-type case (Pusztai, 2017).

A different $N$13-symmetric hyperbolic Ruijsenaars-type system is obtained by Hamiltonian reduction on the Heisenberg double of $N$14. Its reduced Hamiltonian is a three-parameter relativistic many-body system with canonical coordinates $N$15, and its cotangent-bundle limit reproduces the standard three-parameter hyperbolic $N$16 Sutherland Hamiltonian. The construction is explicitly distinct from van Diejen’s $N$17 relativistic models (Marshall, 2013).

There is also a lattice hyperbolic Ruijsenaars model with an exponential Morse term. On the lattice $N$18, its $N$19-particle Hamiltonian is diagonalized by multivariate continuous dual $N$20-Hahn polynomials, obtained as a parameter reduction of Macdonald–Koornwinder polynomials. This produces an exact spectral transform, a bispectral dual $N$21-difference system, a full commuting family of self-adjoint operators, and an $N$22-particle scattering operator whose phase factor decomposes into two-body RS terms and one-body Morse contributions (Diejen et al., 2015).

Several limits connect the hyperbolic model to other integrable systems. In the two-particle setting, the complex rational degeneration $N$23, equivalently $N$24, yields commuting finite-difference operators with rational coefficients in complex variables, Mellin–Barnes and Euler integral representations for eigenfunctions, and complex limits of the hyperbolic Baxter operators. A complementary degeneration occurs at $N$25, or $N$26. The nonrelativistic limit recovers hyperbolic Calogero–Sutherland operators, while additional complex limits produce commuting hypergeometric operators on the cylinder (Belousov et al., 17 Aug 2025).