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

Abstract

The Heun equation can be rewritten as an eigenvalue equation for an ordinary differential operator of the form $-d^2/d{x}^2+V(g;x)$, where the potential is an elliptic function depending on a coupling vector $g\in {\mathbb{R}}^4$. Alternatively, this operator arises from the $B{C}_1$ specialization of the $B{C}_N$ 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 $\mathcal{H}=L^2([0,{\omega }_1],dx)$, where $2{\omega }_1$ is the real period of $V(g;x)$. For this association and a further analysis of $H(g)$, a certain Hilbert-Schmidt operator $\mathcal{I}(g)$ on $\mathcal{H}$ plays a critical role. In particular, using the intimate relation of $H(g)$ and $\mathcal{I}(g)$, we obtain a remarkable spectral invariance: In terms of a coupling vector $c\in {\mathbb{R}}^4$ that depends linearly on $g$, the spectrum of $H(g(c))$ is invariant under arbitrary permutations $\sigma (c)$, $\sigma \in S_4$.

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