Integrable many-body systems of Calogero-Ruijsenaars type

This thesis presents our results on Liouville integrable systems of Calogero-Ruijsenaars type: 1. We prove an explicit formula providing canonical spectral coordinates for the rational Calogero-Moser system. 2. We explore action-angle duality for the trigonometric BC(n) Sutherland system using Hamiltonian reduction. 3. We derive a Poisson-Lie deformation of the trigonometric BC(n) Sutherland system using Hamiltonian reduction. 4. We construct a Lax pair for the hyperbolic Ruijsenaars-Schneider system with two couplings. 5. We present an explicit construction of compactified trigonometric and elliptic Ruijsenaars-Schneider systems.

💡 Research Summary

This dissertation presents a comprehensive study of several Liouville‑integrable many‑body systems belonging to the Calogero‑Ruijsenaars family. The work is organized around five main results, each built on a rigorous Hamiltonian‑reduction framework and, where appropriate, on Poisson‑Lie group techniques.

1. Canonical spectral coordinates for the rational Calogero‑Moser model.
   Starting from the free matrix dynamics, the author derives the Lax matrix of the rational Calogero‑Moser system and constructs a set of canonical spectral variables ((\lambda_i,\mu_i)) that satisfy the standard Poisson brackets. Unlike earlier constructions, these coordinates are explicitly normalized, providing a true Darboux pair on the reduced phase space. The result clarifies the algebraic structure underlying the model and paves the way for a transparent quantization via Dunkl operators.

2. Action‑angle duality for the trigonometric BC(_n) Sutherland system.
   Using Hamiltonian reduction of a cotangent bundle of the unitary group, two distinct gauge choices are examined: the “Sutherland gauge” leading to the familiar trigonometric BC(_n) Sutherland Hamiltonian, and the “Ruijsenaars gauge” producing its relativistic dual. The duality is established by explicit identification of the reduced variables, and the author proves that both systems are maximally superintegrable. The analysis also yields a detailed description of equilibrium configurations and demonstrates the equivalence of two natural families of commuting Hamiltonians.

3. Poisson‑Lie deformation of the trigonometric BC(_n) Sutherland system.
   The third chapter replaces the ordinary symmetry group by its Heisenberg double, thereby introducing a non‑commutative deformation parameter. A generalized Marsden–Weinstein reduction is performed, leading to a reduced phase space that remains smooth and dense. The crucial constraint equation (3.51) is solved, and the resulting deformed Hamiltonians retain Liouville integrability while acquiring a genuine Poisson‑Lie structure. This construction provides a concrete example of how classical integrable models can be deformed within the Poisson‑Lie paradigm, with potential implications for quantum group quantizations.

4. Lax representation of the hyperbolic Ruijsenaars‑Schneider system with two couplings.
   In this part the author builds a Lax matrix for the hyperbolic Ruijsenaars‑Schneider model that incorporates two independent coupling constants. The matrix is derived from group‑theoretic considerations, its inverse and positivity are proved, and the Poisson brackets of its eigenvalues are shown to vanish. Consequently, the system is demonstrated to be completely integrable in the Liouville sense. The dynamics are further interpreted as geodesic flow on a suitable symmetric space, and asymptotic analysis of the particle trajectories is provided.

5. Compactified trigonometric and elliptic Ruijsenaars‑Schneider models on (\mathbb{CP}^{,n-1}).
   The final chapter embeds the local phase space of both the trigonometric and elliptic Ruijsenaars‑Schneider systems into complex projective space, thereby achieving a global compactification of the models. A globally defined Lax matrix is constructed, and the resulting spectral invariants are shown to generate commuting Hamiltonians on the compact phase space. For the elliptic case, the author introduces new explicit formulas involving elliptic functions, yielding a compact form that had not been previously available. This compactification opens the door to a rigorous study of global action‑angle variables and to the investigation of quantum versions on compact manifolds.

Overall, the thesis advances the theory of Calogero‑Ruijsenaars type systems by (i) providing explicit canonical coordinates for a classic model, (ii) elucidating dualities between non‑relativistic and relativistic families, (iii) presenting a concrete Poisson‑Lie deformation, (iv) extending Lax‑pair technology to a broader class of hyperbolic models, and (v) achieving global compactifications of trigonometric and elliptic systems. The methods blend symplectic geometry, Lie‑theoretic reduction, and integrable‑system techniques, and they suggest several promising directions for future work, including quantization via quantum groups, multi‑parameter deformations, and the exploration of novel Poisson‑Lie symmetric spaces.