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

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 ∈ R 4. Alternatively, this operator arises from the BC 1 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 = L 2 ([0, ω 1 ], dx), where 2ω 1 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 ∈ R 4 that depends linearly on g, the spectrum of H(g(c)) is invariant under arbitrary permutations σ(c), σ ∈ S 4 .