Pantheon SEMPARIS Le serveur des séminaires parisiens Paris

Status Confirmed
Seminar Series WORK-CONF
Subjects math-ph
Date Friday 18 August 2017
Time 15:30
Institute LPTENS
Seminar Room Room Conf. IV
Speaker's Last Name Ruijsenaars
Speaker's First Name Simon
Speaker's Email Address
Speaker's Institution University of Leeds
Title Relativistic Heun equation and their $E_8$ spectral invariance
Abstract 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.
arXiv Preprint Number
Comments Workshop on "Exceptional and ubiquitous Painlevé equations for Physics". Please see webpage

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