Statut  Confirmé 
Série  WORKCONF 
Domaines  mathph 
Date  Vendredi 18 Août 2017 
Heure  15:30 
Institut  LPTENS 
Salle  Room Conf. IV 
Nom de l'orateur  Ruijsenaars 
Prenom de l'orateur  Simon 
Addresse email de l'orateur  
Institution de l'orateur  University of Leeds 
Titre  Relativistic Heun equation and their $E_8$ spectral invariance 
Résumé  The eigenvalue equation for the Hamiltonian defining the nonrelativistic quantum elliptic $BC_1$ Calogero Moser system is equivalent to the Heun equation. This linear 4parameter differential equation is closely connected to the nonlinear 4parameter Painlevé VI equation, and the connection persists at lower levels of the two hierarchies. Decades ago, van Diejen introduced an 8parameter difference equation generalizing the Heun equation. It may be viewed as the eigenvalue equation for the Hamiltonian defining the relativistic quantum elliptic $BC_1$ CalogeroMoser system. We sketch our recent results concerning the $E_8$ spectral invariance of a Hilbert space version of this difference operator. This selfadjoint version yields a commuting selfadjoint `modular partner’ with a discrete spectrum that is also invariant under the $E_8$ Weyl group. Our findings are a strong indication of a connection to Sakai’s highest level elliptic difference Painlevé equation, which also has $E_8$ symmetry. At lower levels in the two hierarchies, recent results by Takemura have strengthened this connection. He has shown that the linear Lax equations for the Painlevé difference equations studied by Jimbo / Sakai and Yamada can be tied in with special cases of van Diejen’s relativistic Heun equation. 
Numéro de preprint arXiv  
Commentaires  Workshop on "Exceptional and ubiquitous Painlevé equations for Physics". Please see webpage https://indico.in2p3.fr/event/14720/ 
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