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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. |
