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

article

Simon N.M. Ruijsenaars¹ 

[1] School of Mathematics, University of Leeds

关键词: relativistic Heun equation;    van Diejen operator;    Hilbert–Schmidt operators;    isospectrality;    spectral asymptotics;   

DOI  :  10.3842/SIGMA.2015.004

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown