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Integrable $$BC_N$$ analytic difference operators: hidden parameter symmetries and eigenfunctions. (English) Zbl 1061.35094
Shabat, A.B.(ed.) et al., New trends in integrability and partial solvability. Proceedings of the NATO Advanced Research Workshop, Cadiz, Spain, June 12–16, 2002. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1835-5/hbk). NATO Science Series II: Mathematics, Physics and Chemistry 132, 217-261 (2004).
Summary: We consider integrable $$N$$-particle quantum systems of Calogero-Moser type, focusing on the ‘relativistic’ $$BC_N$$ setting, where commuting analytic difference operators arise. We show that the defining operators at the hyperbolic/elliptic levels, which depend on four/eight coupling constants, can be transformed to a manifestly $$D_4/D_8$$ symmetric form, respectively We survey various results on special eigenfunctions (including ‘ground states’) with regard to the latter symmetries and other ones. We also sketch a symmetry scenario for the arbitrary-$$N$$ eigenfunctions, motivated by the hyperbolic $$BC_1$$ case, where our ‘relativistic’ hypergeometric function has all of the expected properties.
For the entire collection see [Zbl 1050.35003].

##### MSC:
 35Q40 PDEs in connection with quantum mechanics 35Q75 PDEs in connection with relativity and gravitational theory 37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions 81R12 Groups and algebras in quantum theory and relations with integrable systems