Pantheon SEMPARIS Le serveur des séminaires parisiens Paris

Statut Confirmé
Domaines math-ph
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 4-parameter differential equation is closely connected to the nonlinear 4-parameter Painlevé VI equation, and the connection persists at lower levels of the two hierarchies. Decades ago, van Diejen introduced an 8-parameter difference equation generalizing the Heun equation. It may be viewed as the eigenvalue equation for the Hamiltonian defining the relativistic quantum elliptic $BC_1$ Calogero-Moser system. We sketch our recent results concerning the $E_8$ spectral invariance of a Hilbert space version of this difference operator. This self-adjoint version yields a commuting self-adjoint `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
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