Status  Confirmed 
Seminar Series  WORKCONF 
Subjects  mathph 
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 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. 
arXiv Preprint Number  
Comments  Workshop on "Exceptional and ubiquitous Painlevé equations for Physics". Please see webpage https://indico.in2p3.fr/event/14720/ 
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