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

Simon N. M. Ruijsenaars

Symmetry, integrability and geometry: methods and applications, Tome 5 (2009)

Résumé

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 $g\in\mathbb R^4$. 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 $\mathcal H=L^2([0,\omega_1],\,dx)$, 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 $\sigma(c)$, $\sigma\in S_4$.

Détail

Keywords: Heun equation; Hilbert–Schmidt operators; spectral invariance.

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