arXiv subjects

Simon N. M. Ruijsenaars

Publications and source records attributed to Simon N. M. Ruijsenaars.

4 recordsLinked to original sources

arXiv2015

Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case

The 'relativistic' Heun equation is an 8-coupling difference equation that generalizes the 4-coupling Heun differential equation. It can be viewed as the time-independent Schrödinger equation for an analytic difference operator introduced by van Diejen. We study Hilbert space features of this operator and its 'modular partner', based on an in-depth analysis of the eigenvectors of a Hilbert-Schmidt integral operator whose integral kernel has a previously known relation to the two difference operators. With suitable restrictions on the parameters, we show that the commuting difference operators can be promoted to a modular pair of self-adjoint commuting operators, which share their eigenvectors with the integral operator. Various remarkable spectral symmetries and commutativity properties follow from this correspondence. In particular, with couplings varying over a suitable ball in ${\mathbb R}^8$, the discrete spectra of the operator pair are invariant under the $E_8$ Weyl group. The asymptotic behavior of an 8-parameter family of orthonormal polynomials is shown to be shared by the joint eigenvectors.

math-ph↗

arXiv2009

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

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 ${\mathcal H}=L^2([0,ω_1],dx)$, 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 ${\mathcal I}(g)$ 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 $σ(c)$, $σ\in S_4$.

math-ph↗

arXiv2001

Reflectionless analytic difference operators II. Relations to soliton systems

This is the second part of a series of papers dealing with an extensive class of analytic difference operators admitting reflectionless eigenfunctions. In the first part, the pertinent difference operators and their reflectionless eigenfunctions are constructed from given ``spectral data'', in analogy with the IST for reflectionless Schrödinger and Jacobi operators. In the present paper, we introduce a suitable time dependence in the data, arriving at explicit solutions to a nonlocal evolution equation of Toda type, which may be viewed as an analog of the KdV and Toda lattice equations for the latter operators. As a corollary, we reobtain various known results concerning reflectionless Schrödinger and Jacobi operators. Exploiting a reparametrization in terms of relativistic Calogero--Moser systems, we also present a detailed study of $N$-soliton solutions to our nonlocal evolution equation.

nlin.SI↗

arXiv2001

Reflectionless analytic difference operators I. algebraic framework

We introduce and study a class of analytic difference operators admitting reflectionless eigenfunctions. Our construction of the class is patterned after the Inverse Scattering Transform for the reflectionless self-adjoint Schrödinger and Jacobi operators corresponding to KdV and Toda lattice solitons.

nlin.SI↗