480 research outputs found

### 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
${\mathcal H}=L^2([0,\omega_1],dx)$, 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 $\sigma(c)$, $\sigma\in
S_4$

### 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 Schr\"odinger 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 Schr\"odinger 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

### Joint eigenfunctions for the relativistic Calogero-Moser Hamiltonians of hyperbolic type. III. Factorized asymptotics

In the two preceding parts of this series of papers, we introduced and
studied a recursion scheme for constructing joint eigenfunctions $J_N(a_+,
a_-,b;x,y)$ of the Hamiltonians arising in the integrable $N$-particle systems
of hyperbolic relativistic Calogero-Moser type. We focused on the first steps
of the scheme in Part I, and on the cases $N=2$ and $N=3$ in Part II. In this
paper, we determine the dominant asymptotics of a similarity transformed
function \rE_N(b;x,y) for $y_j-y_{j+1}\to\infty$, $j=1,\ldots, N-1$, and
thereby confirm the long standing conjecture that the particles in the
hyperbolic relativistic Calogero-Moser system exhibit soliton scattering. This
result generalizes a main result in Part II to all particle numbers $N>3$.Comment: 21 page

### Self-duality of the compactified Ruijsenaars-Schneider system from quasi-Hamiltonian reduction

The Delzant theorem of symplectic topology is used to derive the completely
integrable compactified Ruijsenaars-Schneider III(b) system from a
quasi-Hamiltonian reduction of the internally fused double SU(n) x SU(n). In
particular, the reduced spectral functions depending respectively on the first
and second SU(n) factor of the double engender two toric moment maps on the
III(b) phase space CP(n-1) that play the roles of action-variables and
particle-positions. A suitable central extension of the SL(2,Z) mapping class
group of the torus with one boundary component is shown to act on the
quasi-Hamiltonian double by automorphisms and, upon reduction, the standard
generator S of the mapping class group is proved to descend to the Ruijsenaars
self-duality symplectomorphism that exchanges the toric moment maps. We give
also two new presentations of this duality map: one as the composition of two
Delzant symplectomorphisms and the other as the composition of three Dehn twist
symplectomorphisms realized by Goldman twist flows. Through the well-known
relation between quasi-Hamiltonian manifolds and moduli spaces, our results
rigorously establish the validity of the interpretation [going back to Gorsky
and Nekrasov] of the III(b) system in terms of flat SU(n) connections on the
one-holed torus.Comment: Final version to appear in Nuclear Physics B, with simplified proof
of Theorem 1, 56 page

### Kernel functions and B\"acklund transformations for relativistic Calogero-Moser and Toda systems

We obtain kernel functions associated with the quantum relativistic Toda
systems, both for the periodic version and for the nonperiodic version with its
dual. This involves taking limits of previously known results concerning kernel
functions for the elliptic and hyperbolic relativistic Calogero-Moser systems.
We show that the special kernel functions at issue admit a limit that yields
generating functions of B\"acklund transformations for the classical
relativistic Calogero-Moser and Toda systems. We also obtain the
nonrelativistic counterparts of our results, which tie in with previous results
in the literature.Comment: 76 page

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