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doi:10.3842/SIGMA.2009.049 Abstract. The Heun equation can be rewritten as an eigenvalue equation for an ordinary differential operator of the form −d 2 /dx 2 + 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 BCN 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 2000 Mathematics Subject Classification: 33E05; 33E10; 46N50; 81Q05; 81Q1

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