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arXiv:2305.18168

One-parameter discrete-time Calogero-Moser system

Umpon Pon Jairuk, Sikarin Yoo-Kong

Published May 29, 2023• 0 citations
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Abstract

We present a new type of integrable one-dimensional many-body systems called a one-parameter Calogero-Moser (CM) system. In the discrete level, the Lax pairs with a parameter are introduced and, of course, the discrete-time equations of motion are obtained as well as their corresponding discrete-time Lagrangian. The integrability feature of this new system can be captured through the discrete Lagrangian closure relation by employing a connection with the temporal Lax matrices of the discrete-time Ruijsenaars-Schneider (RS) system, exact solution, and the existence of the classical r-matrix. Under the appropriate limit on the parameter, which in this case is approaching zero, the standard CM system is retrieved both discrete-time and continuous-time.

DOI
10.48550/arxiv.2305.18168

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We present a new type of integrable one-dimensional many-body systems called a one-parameter Calogero-Moser (CM) system. In the discrete level, the Lax pairs with a parameter are introduced and, of course, the discrete-time equations of motion are obtained as well as their corresponding discrete-time Lagrangian. The integrability feature of this new system can be captured through the discrete Lagrangian closure relation by employing a connection with the temporal Lax matrices of the discrete-time Ruijsenaars-Schneider (RS) system, exact solution, and the existence of the classical r-matrix. Under the appropriate limit on the parameter, which in this case is approaching zero, the standard CM system is retrieved both discrete-time and continuous-time.
The review of the results in the theory of integrable many-body systems disconnected with semisimple Lie algebras is done. The one-dimensional systems of light Calogero–Sutherland–Moser particles interacting with one particle of infinite mass located at the origin are described in detail. In some cases the exact solutions of the equations of motion are obtained. The general theory of integration of the equations of motion needs the methods of algebraic geometry. The Lax pairs with spectral parameter are constructed for this purpose. The theory still contains many unsolved problems.
Recent interest in discrete, classical integrable systems has focused on their connection to quantum integrable systems via the Bethe equations. In this note, solutions to the rational nested Bethe ansatz (RNBA) equations are constructed using the ``completed Calogero-Moser phase space'' of matrices which satisfy a finite dimensional analogue of the canonical commutation relationship. A key feature is the fact that the RNBA equations are derived only from this commutation relationship and some elementary linear algebra. The solutions constructed in this way inherit continuous and discrete symmetries from the CM phase space.
The expression of the quantum Ruijsenaars-Schneider Hamiltonian is obtained in the framework of the dynamical RRR-matrix formalism. This generalizes to the case of Uq(sln)U_q(sl_n)Uq​(sln​) the result obtained by O. Babelon, D. Bernard and E. Billey for Uq(sl2)U_q(sl_2)Uq​(sl2​) which is the higher difference Lame operator. The general method involved is the universal RRR-matrix construction.