|
This article is cited in 8 scientific papers (total in 8 papers)
Hilbert–Schmidt Operators vs. Integrable Systems of Elliptic Calogero–Moser Type. III. The Heun Case
Simon N. M. Ruijsenaarsab a Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
b Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
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 $\sigma(c)$, $\sigma\in S_4$.
Keywords:
Heun equation; Hilbert–Schmidt operators; spectral invariance.
Received: January 19, 2009; Published online April 21, 2009
Citation:
Simon N. M. Ruijsenaars, “Hilbert–Schmidt Operators vs. Integrable Systems of Elliptic Calogero–Moser Type. III. The Heun Case”, SIGMA, 5 (2009), 049, 21 pp.
Linking options:
https://www.mathnet.ru/eng/sigma395 https://www.mathnet.ru/eng/sigma/v5/p49
|
Statistics & downloads: |
Abstract page: | 163 | Full-text PDF : | 39 | References: | 55 |
|