Geodesic
Hilbert–Schmidt Operators vs. Integrable Systems of Elliptic Calogero–Moser Type IV. The Relativistic Heun (van Diejen) Case
Symmetry, integrability and geometry: methods and applications, Tome 11 (2015) Cet article a éte moissonné depuis la source Math-Net.Ru

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The ‘relativistic’ Heun equation is an 8-coupling difference equation that generalizes the 4-coupling Heun differential equation. It can be viewed as the time-independent Schrödinger equation for an analytic difference operator introduced by van Diejen. We study Hilbert space features of this operator and its ‘modular partner’, based on an in-depth analysis of the eigenvectors of a Hilbert–Schmidt integral operator whose integral kernel has a previously known relation to the two difference operators. With suitable restrictions on the parameters, we show that the commuting difference operators can be promoted to a modular pair of self-adjoint commuting operators, which share their eigenvectors with the integral operator. Various remarkable spectral symmetries and commutativity properties follow from this correspondence. In particular, with couplings varying over a suitable ball in ${\mathbb R}^8$, the discrete spectra of the operator pair are invariant under the $E_8$ Weyl group. The asymptotic behavior of an 8-parameter family of orthonormal polynomials is shown to be shared by the joint eigenvectors.
Keywords: relativistic Heun equation; van Diejen operator; Hilbert–Schmidt operators; isospectrality; spectral asymptotics. 
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     author = {Simon N. M. Ruijsenaars},
     title = {Hilbert{\textendash}Schmidt {Operators} {vs.~Integrable} {Systems} of {Elliptic} {Calogero{\textendash}Moser} {Type~IV.} {The} {Relativistic} {Heun} (van {Diejen)} {Case}},
     journal = {Symmetry, integrability and geometry: methods and applications},
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Simon N. M. Ruijsenaars. Hilbert–Schmidt Operators vs. Integrable Systems of Elliptic Calogero–Moser Type IV. The Relativistic Heun (van Diejen) Case. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a3/

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