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Construction of eigenfunctions for the elliptic Ruijsenaars difference operators. (English) Zbl 1495.81062

Summary: We present a perturbative construction of two kinds of eigenfunctions of the commuting family of difference operators defining the elliptic Ruijsenaars system. The first kind corresponds to elliptic deformations of the Macdonald polynomials, and the second kind generalizes asymptotically free eigenfunctions previously constructed in the trigonometric case. We obtain these eigenfunctions as infinite series which, as we show, converge in suitable domains of the variables and parameters. Our results imply that, for the domain where the elliptic Ruijsenaars operators define a relativistic quantum mechanical system, the elliptic deformations of the Macdonald polynomials provide a family of orthogonal functions with respect to the pertinent scalar product.

MSC:

81R12 Groups and algebras in quantum theory and relations with integrable systems
33D52 Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
39A70 Difference operators
81V70 Many-body theory; quantum Hall effect
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