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

TL;DR

This paper explores the spectral properties of a differential operator linked to the Heun equation and elliptic Calogero-Moser systems, revealing a spectral invariance under permutations of coupling parameters.

Contribution

It establishes a connection between the Heun equation, elliptic Calogero-Moser systems, and Hilbert-Schmidt operators, demonstrating spectral invariance under parameter permutations.

Findings

Spectral invariance under permutation of coupling parameters.

Construction of a self-adjoint operator associated with the Heun equation.

Identification of a Hilbert-Schmidt operator critical to spectral analysis.

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 $g\in{\mathbb R}^4$. 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 $\sigma(c)$, $\sigma\in S_4$.