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Three formulas for eigenfunctions of integrable Schrödinger operators

Published online by Cambridge University Press:  04 December 2007

GIOVANNI FELDER
Affiliation:
Department of Mathematics, Phillips Hall, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA e-mail: felder@math.unc.edu, varchenko@math.unc.edu.
ALEXANDER VARCHENKO
Affiliation:
Department of Mathematics, Phillips Hall, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA e-mail: felder@math.unc.edu, varchenko@math.unc.edu.
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Abstract

We give three formulas for meromorphic eigenfunctions (scattering states) of Sutherland's integrable $N$-body Schrödinger operators and their generalizations. The first is an explicit computation of the Etingof-Kirillov traces of intertwining operators, the second an integral representation of hypergeometric type, and the third is a formula of Bethe ansatz type. The last two formulas are degenerations of elliptic formulas obtained previously in connection with the Knizhnik-Zamolodchikov-Bernard equation. The Bethe ansatz formulas in the elliptic case are reviewed and discussed in more detail here: Eigenfunctions are parametrized by a ‘Hermite-Bethe’ variety, a generalization of the spectral variety of the Lamé operator. We also give the $q$-deformed version of our first formula. In the scalar ${\rm sl}_N$ case, this gives common eigenfunctions of the commuting Macdonald-Rujsenaars difference operators.

Type
Research Article
Copyright
© 1997 Kluwer Academic Publishers

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