• DOI:10.3842/SIGMA.2015.004
• Corpus ID: 119176526

Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case

@article{Ruijsenaars2015HilbertSchmidtOV,
  title={Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case},
  author={Simon N. M. Ruijsenaars},
  journal={Symmetry Integrability and Geometry-methods and Applications},
  year={2015},
  volume={11},
  pages={004}
}

• S. Ruijsenaars
• Published 16 April 2014
• Mathematics
• Symmetry Integrability and Geometry-methods and Applications

The 'relativistic' Heun equation is an 8-coupling difference equation that gene- ralizes the 4-coupling Heun differential equation. It can be viewed as the time-independent Schrodinger 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 R 8 , the discrete spectra of the operator pair are invariant under the E8 Weyl group. The asymptotic behavior of an 8-parameter family of orthonormal polynomials is shown to be shared by the joint eigenvectors.