Geodesic
Baxter $Q$-operators in Ruijsenaars–Sutherland hyperbolic systems: one- and two-particle cases
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 29, Tome 520 (2023), pp. 50-123 Cet article a éte moissonné depuis la source Math-Net.Ru

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In these notes we review the technique of Baxter $Q$-operators in the Ruijsenaars-Sutherland hyperbolic systems in the cases of one and two particles. Using these operators we show in particular that eigenfunctions of these systems admit two dual integral representations and prove their orthogonality and completeness. 
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N. Belousov; S. Derkachov; S. Kharchev; S. Khoroshkin. Baxter $Q$-operators in Ruijsenaars–Sutherland hyperbolic systems: one- and two-particle cases. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 29, Tome 520 (2023), pp. 50-123. http://geodesic.mathdoc.fr/item/ZNSL_2023_520_a2/

[1] E. W. Barnes, “The theory of the double gamma function”, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 196 (1901), 265–387 | MR

[2] N. Belousov, S. Derkachov, S. Kharchev, S. Khoroshkin, Baxter operators in Ruijsenaars hyperbolic system I. Commutativity of $Q$-operators, arXiv: 2303.06383

[3] N. Belousov, S. Derkachov, S. Kharchev, S. Khoroshkin, Baxter operators in Ruijsenaars hyperbolic system II. Bispectral wave functions, arXiv: 2303.06382

[4] L. D. Faddeev, “Discrete Heisenberg–Weyl Group and modular group”, Lett. Math. Phys., 34 (1995), 249–254 | DOI | MR | Zbl

[5] L. D. Faddeev, R. M. Kashaev, A. Yu. Volkov, “Strongly coupled quantum discrete Liouville theory. I: Algebraic approach and duality”, Commun. Math. Phys., 219 (2001), 199–219 | DOI | MR | Zbl

[6] L. D. Faddeev, O. A. Yakubovsky, Lectures in Quantum Mechanics for Mathematician Students, Student Mathematical Library, 47, AMS, 2009 | DOI | MR

[7] I. M. Gelfand, G. E. Shilov, Generalized Functions: Properties and operations, Academic Press, 1964 | MR | Zbl

[8] M. Hallnäs, S. Ruijsenaars, “Joint eigenfunctions for the relativistic Calogero–Moser Hamiltonians of hyperbolic type: I. First steps”, Int. Math. Res. Notices, 2014, 4400–4456 | DOI | MR | Zbl

[9] M. Hallnäs, S. Ruijsenaars, “A recursive construction of joint eigenfunctions for the hyperbolic nonrelativistic Calogero-Moser Hamiltonians”, Int. Math. Res. Notices, 2015, 10278–10313 | DOI | MR | Zbl

[10] M. Hallnäs, S. Ruijsenaars, “Product formulas for the relativistic and nonrelativistic conical functions”, Adv. Stud. Pure Math., 76 (2018), 195–246 | DOI | MR

[11] S. Kharchev, S. Khoroshkin, Wave function for $GL(n,\mathbb{R})$ hyperbolic Sutherland model, arXiv: 2108.04895

[12] N. Kurokawa, S-Y. Koyama, “Multiple sine functions”, Forum Math., 15 (2003), 839–876 | DOI | MR | Zbl

[13] M. L. Nazarov, E. K. Sklyanin, “Sekiguchi–Debiard operators at infinity”, Commun. Math. Phys., 324 (2013), 831–849 | DOI | MR | Zbl

[14] M. L. Nazarov, E. K. Sklyanin, “Macdonald operators at infinity”, J. Algebraic Combin., 40 (2013), 23–44 | DOI | MR

[15] B. Ponsot, J. Teschner, “Clebsch–Gordan and Racah–Wigner coefficients for a continuous series of representations of $ U_q (sl(2, \mathbb{R}))$”, Commun. Mathe. Phys., 224 (2001), 613–655 | DOI | MR

[16] S. N. M. Ruijsenaars, “First-order analytic difference equations and integrable quantum systems”, J. Math. Phys., 38 (1997), 1069–1146 | DOI | MR | Zbl

[17] S. N. M. Ruijsenaars, “Zero-eigenvalue eigenfunctions for differences of elliptic relativistic Calogero–Moser Hamiltonians”, Theor. Math. Phys., 146:1 (2006), 25–33 | DOI | Zbl