HilbertSchmidt Operators vs. Integrable Systems of Elliptic CalogeroMoser Type IV. The Relativistic Heun (van Diejen) Case
Abstract
The 'relativistic' Heun equation is an 8coupling difference equation that generalizes the 4coupling Heun differential equation. It can be viewed as the timeindependent 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 indepth analysis of the eigenvectors of a HilbertSchmidt 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 selfadjoint 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 E_8 Weyl group. The asymptotic behavior of an 8parameter family of orthonormal polynomials is shown to be shared by the joint eigenvectors.
 Publication:

SIGMA
 Pub Date:
 January 2015
 DOI:
 10.3842/SIGMA.2015.004
 arXiv:
 arXiv:1404.4392
 Bibcode:
 2015SIGMA..11..004R
 Keywords:

 relativistic Heun equation;
 van Diejen operator;
 HilbertSchmidt operators;
 isospectrality;
 spectral asymptotics;
 Mathematical Physics;
 Mathematics  Functional Analysis;
 Mathematics  Quantum Algebra;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 Quantum Physics
 EPrint:
 SIGMA 11 (2015), 004, 78 pages