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 Sakais 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 Diejens relativistic Heun equation. |
arXiv Preprint Number | |
Comments | Workshop on "Exceptional and ubiquitous Painlevé equations for Physics". Please see webpage https://indico.in2p3.fr/event/14720/ |
Attachments |
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