Symmetry, Integrability and Geometry: Methods and Applications (Apr 2009)
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/dx^2+V(g;x), where the potential is an elliptic function depending on a coupling vector g in 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 in R4 that depends linearly on g, the spectrum of H(g(c)) is invariant under arbitrary permutations σ(c), σ in S_4.
