Geodesic
Reflection operator and hypergeometry I: $SL(2,\mathbb{R})$ spin chain
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 30, Tome 532 (2024), pp. 5-46 
P. V. Antonenko; N. M. Belousov; S. È. Derkachov; S. M. Khoroshkin. Reflection operator and hypergeometry I: $SL(2,\mathbb{R})$ spin chain. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 30, Tome 532 (2024), pp. 5-46. http://geodesic.mathdoc.fr/item/ZNSL_2024_532_a0/
@article{ZNSL_2024_532_a0,
     author = {P. V. Antonenko and N. M. Belousov and S. \`E. Derkachov and S. M. Khoroshkin},
     title = {Reflection operator and hypergeometry {I:} $SL(2,\mathbb{R})$ spin chain},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--46},
     year = {2024},
     volume = {532},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2024_532_a0/}
}
TY  - JOUR
AU  - P. V. Antonenko
AU  - N. M. Belousov
AU  - S. È. Derkachov
AU  - S. M. Khoroshkin
TI  - Reflection operator and hypergeometry I: $SL(2,\mathbb{R})$ spin chain
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2024
SP  - 5
EP  - 46
VL  - 532
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2024_532_a0/
LA  - en
ID  - ZNSL_2024_532_a0
ER  - 
%0 Journal Article
%A P. V. Antonenko
%A N. M. Belousov
%A S. È. Derkachov
%A S. M. Khoroshkin
%T Reflection operator and hypergeometry I: $SL(2,\mathbb{R})$ spin chain
%J Zapiski Nauchnykh Seminarov POMI
%D 2024
%P 5-46
%V 532
%U http://geodesic.mathdoc.fr/item/ZNSL_2024_532_a0/
%G en
%F ZNSL_2024_532_a0

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

In this work we consider open $SL(2,\mathbb{R})$ spin chain, mainly the simplest case of one particle. Eigenfunctions of the model can be constructed using the so-called reflection operator. We obtain several representations of this operator and show its relation to the hypergeometric function. Besides, we prove orthogonality and completeness of one-particle eigenfunctions and connect them to the index hypergeometric transform. Finally, we briefly state the formula for the eigenfunctions in many-particle case.

[1] G. E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, 1999 | MR | Zbl

[2] P. Antonenko, N. Belousov, S. Derkachov, P. Valinevich, Reflection operator and hypergeometry II: $SL(2, \mathbb{C})$ spin chain, to appear, 2024 | MR

[3] A. V. Belitsky, S. E. Derkachov, A. N. Manashov, “Quantum mechanics of null polygonal Wilson loops”, Nuclear Physics B, 882 (2014), 303–351 | DOI | MR | Zbl

[4] I. V. Cherednik, “Factorizing Particles on a Half Line and Root Systems”, Theor. Math. Phys., 61 (1984), 977–983 | DOI | MR | Zbl

[5] S. E. Derkachov, G. P. Korchemsky, A. N. Manashov, “Separation of variables for the quantum $SL (2, \mathbb{R})$ spin chain”, JHEP, 2003:07 (2003), 047 | DOI | MR

[6] S. E. Derkachov, G. P. Korchemsky, A. N. Manashov, “Baxter $Q$-operator and separation of variables for the open $SL(2,R)$ spin chain”, JHEP, 2003:10 (2003), 053 | DOI | MR

[7] F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain (eds.), NIST Digital Library of Mathematical Functions, Release 1.2.0 of 2024-03-15

[8] R. Frassek, C. Giardinà, J. Kurchan, “Non-compact quantum spin chains as integrable stochastic particle processes”, J. Stat. Physics, 180 (2020), 135–171 | DOI | MR | Zbl

[9] L. D. Faddeev, O. A. Yakubovski\u i, Lectures on Quantum Mechanics for Mathematics Students, American Mathematical Society, 2009 | MR | Zbl

[10] R. A. Gustafson, “Some q-Beta and Mellin-Barnes Integrals on Compact Lie Groups and Lie Algebras”, Transactions of the American Mathematical Society, 341:1 (1994), 69–119 | MR | Zbl

[11] I. M. Gelfand, M. I. Graev, N. Ya. Vilenkin, Generalized functions, v. 5, Integral geometry and representation theory, AMS Chelsea Publishing, 1966 | MR

[12] T. Koornwinder, “A new proof of a Paley–Wiener type theorem for the Jacobi transform”, Ark. Mat., 13 (1975), 145–159 | DOI | MR | Zbl

[13] S. Farid Khwaja, A. B. Olde Daalhuis, “Uniform asymptotic expansions for hypergeometric functions with large parameters IV”, Analysis and Applications, 12:06 (2014), 667–710 | DOI | MR | Zbl

[14] P. P. Kulish, E. K. Sklyanin, “Algebraic structures related to the reflection equations”, J. Phys. A, 25 (1992), 5963–5976 | DOI | MR

[15] Yu. A. Neretin, “Index hypergeometric integral transform”, addendum to the Russian translation of G. E. Andrews, R. Askey, and R. Roy, Special functions, MCCME, M., 2013, 607–624; arXiv: 1208.3342 [math.CA]

[16] R. B. Paris, D. Kaminski, Asymptotics and Mellin-Barnes Integrals, Cambridge University Press, 2001 | MR | Zbl

[17] E. K. Sklyanin, “Boundary Conditions for Integrable Quantum Systems”, J. Phys. A, 21 (1988), 2375–2389 | DOI | MR | Zbl