Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type I. The Eigenfunction Identities

In this series of papers we study Hilbert-Schmidt integral operators acting on the Hilbert spaces associated with elliptic Calogero-Moser type Hamiltonians. As shown in this first part, the integral kernels are joint eigenfunctions of differences of the latter Hamiltonians. On the relativistic (difference operator) level the kernel is built from the elliptic gamma function, whereas the building block in the nonrelativistic (differential operator) limit is basically the Weierstrass sigma-function. For the AN−1 case we consider all of the commuting Hamiltonians at once, the eigenfunction properties reducing to a sequence of elliptic identities. For the BCN case we only treat the defining Hamiltonians. The functional identities encoding the eigenfunction properties have a remarkable corollary in the relativistic BC1 case: They imply that the sum over eight-fold products of the four Jacobi theta functions is invariant under the Weyl group of E8.