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

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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. 
@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},
     publisher = {mathdoc},
     volume = {532},
     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2024_532_a0/}
}
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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/