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Dunkl operators and Calogero-Sutherland models. (English) Zbl 1061.35090
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, 157-189 (2004).
Summary: We describe a general method for constructing (scalar or spin) Calogero-Sutherland models of \(A_N\) or \(BC_N\) type, which are either exactly or quasi-exactly solvable. Our approach is based on the simultaneous use of three different families of Dunkl operators of each type, one of which was recently introduced by the authors. We perform a complete classification of the models which can be constructed by our method. We obtain in this way several new families of (quasi-)exactly solvable Calogero-Sutherland models, some of them with elliptic interactions.
For the entire collection see [Zbl 1050.35003].
35Q40 PDEs in connection with quantum mechanics
81T10 Model quantum field theories
81V70 Many-body theory; quantum Hall effect
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures