Abstract
We consider the problem of finding integrals of motion for quantum elliptic Calogero-Moser systems with arbitrary number of particles extended by introducing spinexchange interaction. By direct calculation, after making certain ansatz, we found first two integrals — quite probably, lowest nontrivial members of the whole commutative ring. This result might be considered as the first step in constructing this ring of the operators which commute with the Hamiltonian of the model.
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References
Calogero, F., The Integrable Many-Body Systems, Lett. Nuovo Cimento, 1975, vol. 13, pp. 411–417.
Krichever, I. M., Elliptic Solutions to the Kadomtsev-Petviashvily Equation and Integrable Systems of Particles, Funktsional. Anal. i Prilozhen., 1980, vol. 14, pp. 45–54 [Funct. Anal. Appl., 1980, vol. 14, pp. 282–290].
Olshanetsky, M.A. and Perelomov, A.M., Quantum Integrable Many-Body Systems Related to Lie Algebras, Phys. Rep., 1983, vol. 94, pp. 313–392.
Inozemtsev, V. I., On the Connection between the One-Dimensional S = 1/2 Heisenberg Chain and Haldane-Shastry Model, J. Stat. Phys., 1990, vol. 59, pp. 1146–1157.
Sutherland, B. and Shastry, B. S., Solution of Some Integrable One-Dimensional Quantum Systems, Phys. Rev. Lett., 1993, vol. 71, pp. 5–8.
Dittrich, J. and Inozemtsev, V. I., On the Structure of Eigenvectors of the Multidimensional Lame Operator, J. Phys. A, 1993, vol. 26, pp. L753–L756.
Inozemtsev, V. I., Invariants of Linear Combinations of Transpositions, Lett. Math. Phys., 1996, vol. 36, pp. 55–63.
Dittrich, J. and Inozemtsev, V. I., The Commutativity of Integrals of Motion for Quantum Spin Chains and Elliptic Functions Identities, Regul. Chaotic Dyn., 2008, vol. 13, pp. 19–26.
Barba, J. C. and Inozemtsev, V. I., On the Solutions of 3-Particle Spin Elliptic Calogero-Moser Systems, Phys. Lett. A, 2008, vol. 372, pp. 5951–5954.
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Dittrich, J., Inozemtsev, V.I. Towards the proof of complete integrability of quantum elliptic many-body systems with spin degrees of freedom. Regul. Chaot. Dyn. 14, 218–222 (2009). https://doi.org/10.1134/S1560354709020026
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DOI: https://doi.org/10.1134/S1560354709020026