Geodesic
Unitary Representations of the Quantum Lorentz Group and Quantum Relativistic Toda Chain
Teoretičeskaâ i matematičeskaâ fizika, Tome 130 (2002) no. 3, pp. 355-382

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We give a group theory interpretation of the three types of $q$-Bessel functions. We consider a family of quantum Lorentz groups and a family of quantum Lobachevsky spaces. For three values of the parameter of the quantum Lobachevsky space, the Casimir operators correspond to the two-body relativistic open Toda-chain Hamiltonians whose eigenfunctions are the modified $q$-Bessel functions of the three types. We construct the principal series of unitary irreducible representations of the quantum Lorentz groups. Special matrix elements in the irreducible spaces given by the $q$-Macdonald functions are the wave functions of the two-body relativistic open Toda chain. We obtain integral representations for these functions. 
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     author = {M. A. Olshanetsky and V.-B. K. Rogov},
     title = {Unitary {Representations} of the {Quantum} {Lorentz} {Group} and {Quantum} {Relativistic} {Toda} {Chain}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     url = {http://geodesic.mathdoc.fr/item/TMF_2002_130_3_a0/}
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M. A. Olshanetsky; V.-B. K. Rogov. Unitary Representations of the Quantum Lorentz Group and Quantum Relativistic Toda Chain. Teoretičeskaâ i matematičeskaâ fizika, Tome 130 (2002) no. 3, pp. 355-382. http://geodesic.mathdoc.fr/item/TMF_2002_130_3_a0/