SIGMA 5 (2009), 049, 21 pages      arXiv:0904.3250      https://doi.org/10.3842/SIGMA.2009.049
Contribution to the Proceedings of the Workshop “Elliptic Integrable Systems, Isomonodromy Problems, and Hypergeometric Functions”

Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type III. The Heun Case

Simon N.M. Ruijsenaars ^{a, b
a)} Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
^b) Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK

Received January 19, 2009; Published online April 21, 2009

Abstract
The Heun equation can be rewritten as an eigenvalue equation for an ordinary differential operator of the form −d²/dx²+V(g;x), where the potential is an elliptic function depending on a coupling vector g ∈ R⁴. Alternatively, this operator arises from the BC₁ specialization of the BC_N elliptic nonrelativistic Calogero-Moser system (a.k.a. the Inozemtsev system). Under suitable restrictions on the elliptic periods and on g, we associate to this operator a self-adjoint operator H(g) on the Hilbert space H = L²([0,ω₁],dx), where 2ω₁ is the real period of V(g;x). For this association and a further analysis of H(g), a certain Hilbert-Schmidt operator I(g) on H plays a critical role. In particular, using the intimate relation of H(g) and I(g), we obtain a remarkable spectral invariance: In terms of a coupling vector c ∈ R⁴ that depends linearly on g, the spectrum of H(g(c)) is invariant under arbitrary permutations σ(c), σ ∈ S₄.

Key words: Heun equation; Hilbert-Schmidt operators; spectral invariance.

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