480 research outputs found

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

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    The Heun equation can be rewritten as an eigenvalue equation for an ordinary differential operator of the form d2/dx2+V(g;x)-d^2/dx^2+V(g;x), where the potential is an elliptic function depending on a coupling vector gR4g\in{\mathbb R}^4. Alternatively, this operator arises from the BC1BC_1 specialization of the BCNBC_N elliptic nonrelativistic Calogero-Moser system (a.k.a. the Inozemtsev system). Under suitable restrictions on the elliptic periods and on gg, we associate to this operator a self-adjoint operator H(g)H(g) on the Hilbert space H=L2([0,ω1],dx){\mathcal H}=L^2([0,\omega_1],dx), where 2ω12\omega_1 is the real period of V(g;x)V(g;x). For this association and a further analysis of H(g)H(g), a certain Hilbert-Schmidt operator I(g){\mathcal I}(g) on H{\mathcal H} plays a critical role. In particular, using the intimate relation of H(g)H(g) and I(g){\mathcal I}(g), we obtain a remarkable spectral invariance: In terms of a coupling vector cR4c\in{\mathbb R}^4 that depends linearly on gg, the spectrum of H(g(c))H(g(c)) is invariant under arbitrary permutations σ(c)\sigma(c), σS4\sigma\in S_4

    Reflectionless analytic difference operators II. Relations to soliton systems

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    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 NN-soliton solutions to our nonlocal evolution equation

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

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    In the two preceding parts of this series of papers, we introduced and studied a recursion scheme for constructing joint eigenfunctions JN(a+,a,b;x,y)J_N(a_+, a_-,b;x,y) of the Hamiltonians arising in the integrable NN-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=2N=2 and N=3N=3 in Part II. In this paper, we determine the dominant asymptotics of a similarity transformed function \rE_N(b;x,y) for yjyj+1y_j-y_{j+1}\to\infty, j=1,,N1j=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>3N>3.Comment: 21 page

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

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    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

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    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