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The problem of integrable discretization: Hamiltonian approach
 Progress in Mathematics, Volume 219. Birkhäuser
"... this paper. Hence, the nature of this discretization as a member of the Toda hierarchy was not understood properly. A complete account was given in Suris (1995). The work Rutishauser (1954) contains not only the denition of the qd algorithm from the viewpoint of the numerical analist, but also the e ..."
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this paper. Hence, the nature of this discretization as a member of the Toda hierarchy was not understood properly. A complete account was given in Suris (1995). The work Rutishauser (1954) contains not only the denition of the qd algorithm from the viewpoint of the numerical analist, but also the equations of motion of the Toda lattice (under the name of a \continuous analogue of the qd algorithm")! The relation of the qd algorithm to integrable systems might have important implications for the numerical analysis, cf. Deift et al. (1991), Nagai and Satsuma (1995).
Elementary Toda orbits and integrable lattices
 J. Math. Phys
, 2000
"... We show that key features of several important integrable lattices appear naturally in a framework of the full Toda ows. Using special symplectic leaves for these ows, we construct a family of biHamiltonian integrable lattices that interpolates between the nonrelativistic and relativistic Toda lat ..."
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We show that key features of several important integrable lattices appear naturally in a framework of the full Toda ows. Using special symplectic leaves for these ows, we construct a family of biHamiltonian integrable lattices that interpolates between the nonrelativistic and relativistic Toda lattices. I. Introduction In the study of soliton lattice equations there are two clearly distinguishable trends. On the one hand, there is a tremendous amount of papers devoted to individual integrable lattices or, more generally, hierarchies of dierentialdierence equations, such as the Toda lattice, the Volterra lattice, AblowitzLadik hierarchy etc. More recently, an equal amount of attention has been attracted by the relativistic Toda lattice, introduced in [1]. For a comprehensive account on the latter system we refer the reader to [2]. In this context, it is important to investigate relations between dierent hierarchies, e.g. Backlund transformations of one system into another. Ex...
The general analytic solution of a functional equation of addition type
 Siam J. Math. anal
, 1997
"... functan/9508002 ..."
V.Rogov, Liouville quantum mechanics on a lattice from geometry of quantum Lorentz group
 Journ.Phys.: Math.Gen
, 1994
"... We consider the quantum Lobachevsky space L3 q, which is defined as subalgebra of the Hopf algebra Aq(SL2(C)). The Iwasawa decomposition of Aq(SL2(C)) introduced by Podles and Woronowicz allows to consider the quantum analog of the horospheric coordinates on L3 q. The action of the Casimir element, ..."
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We consider the quantum Lobachevsky space L3 q, which is defined as subalgebra of the Hopf algebra Aq(SL2(C)). The Iwasawa decomposition of Aq(SL2(C)) introduced by Podles and Woronowicz allows to consider the quantum analog of the horospheric coordinates on L3 q. The action of the Casimir element, which belongs to the dual to Aq quantum group Uq(SL2(C)), on some subspace in L3 q in these coordinates leads to a second order difference operator on the infinite onedimensional lattice. In the continuos limit q → 1 it is transformed into the Schrödinger Hamiltonian, which describes zero modes into the Liouville field theory (the Liouville quantum mechanics). We calculate the spectrum (Brillouin zones) and the eigenfunctions of this operator. They are qcontinuos Hermit polynomials, which are particular case of the Macdonald or RogersAskeyIsmail polynomials. The scattering in this problem corresponds to the scattering of first two level dressed excitations in the ZN Baxter model in the very peculiar limit when the anisotropy parameter γ and N → ∞, or, equivalently,
Exact results for topological strings on resolved Y(p,q) singularities,” arXiv:0804.2598 [hepth
"... We obtain exact results in α ′ for open and closed A−model topological string amplitudes on a large class of toric CalabiYau threefolds by using their correspondence with five dimensional gauge theories. The toric CalabiYau’s that we analyze are obtained as minimal resolution of cones over Y p,q s ..."
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We obtain exact results in α ′ for open and closed A−model topological string amplitudes on a large class of toric CalabiYau threefolds by using their correspondence with five dimensional gauge theories. The toric CalabiYau’s that we analyze are obtained as minimal resolution of cones over Y p,q singularities and give rise via Mtheory compactification to SU(p) gauge theories on R 4 × S 1. As an application we present a detailed study of the local F2 case and compute genus zero GromovWitten invariants on the C 3 /Z4 orbifold. The mirror curve in this case is the spectral curve of the relativistic A1 Toda chain. Our results also indicate the existence of a wider class of relativistic integrable systems associated to generic Y p,q geometries.
Integrable Systems with Pairwise Interactions and Functional Equations
 Reviews in Mathematics and Mathematical Physics, vol 10 pt2
, 1997
"... A new ansatz is presented for a Lax pair describing systems of particles on the line interacting via (possibly nonsymmetric) pairwise forces. Particular cases of this yield the known Lax pairs for the CalogeroMoser and Toda systems, as well as their relativistic generalisations. The ansatz leads to ..."
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A new ansatz is presented for a Lax pair describing systems of particles on the line interacting via (possibly nonsymmetric) pairwise forces. Particular cases of this yield the known Lax pairs for the CalogeroMoser and Toda systems, as well as their relativistic generalisations. The ansatz leads to a system of functional equations. Several new functional equations are described and the general analytic solution to some of these is Completely integrable systems arise in various diverse settings in both mathematics and physics and accordingly have been studied from many different points of view, a fact which underlies their importance and interest (see for example [23].) Within this area the study of Lax pairs, a zero curvature condition, plays
Asymptotics of Laurent Polynomials of Even Degree Orthogonal with Respect to Varying Exponential Weights
, 2006
"... LetΛR denote the linear space overRspanned by zk, k∈Z. Define the real inner product (with varying exponential weights)〈·,·〉L:ΛR×Λ R→R, ( f, g)↦ → ∫ f (s)g(s) exp(−N V(s)) ds, N∈N, where R the external field V satisfies: (i) V is real analytic onR\{0}; (ii) limx→∞(V(x) / ln(x2 +1))=+∞; and (iii) ..."
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LetΛR denote the linear space overRspanned by zk, k∈Z. Define the real inner product (with varying exponential weights)〈·,·〉L:ΛR×Λ R→R, ( f, g)↦ → ∫ f (s)g(s) exp(−N V(s)) ds, N∈N, where R the external field V satisfies: (i) V is real analytic onR\{0}; (ii) limx→∞(V(x) / ln(x2 +1))=+∞; and (iii) limx→0(V(x) / ln(x−2 +1))=+∞. Orthogonalisation of the (ordered) base{1, z−1, z, z−2, z2,...,z −k, zk,...} with respect to〈·,·〉L yields the even degree and odd degree orthonormal Laurent polynomials {φm(z)} ∞ (2n):φ2n(z)=ξ m=0 −n z−n +···+ξ (2n) n zn,ξ (2n) n>0, andφ2n+1(z)=ξ (2n+1) −n−1 z−n−1 +···+ξ (2n+1) n zn,ξ (2n+1)>0. Define
ITEPTH40/01 RELATIVISTIC TODA CHAIN AT ROOT OF UNITY III. RELATIVISTIC TODA CHAIN HIERARCHY
, 2001
"... Abstract. The hierarchy of the classical nonlinear integrable equations associated with relativistic Toda chain model is considered. It is formulated for the Npowers of the quantum operators of the corresponding quantum integrable models. Following the ideas of the paper [11] it is shown how one ca ..."
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Abstract. The hierarchy of the classical nonlinear integrable equations associated with relativistic Toda chain model is considered. It is formulated for the Npowers of the quantum operators of the corresponding quantum integrable models. Following the ideas of the paper [11] it is shown how one can obtain such a system from 2D Toda lattice system. The reduction procedure is described explicitly. The soliton solutions for the relativistic Toda chain are constructed using results of [12] in terms of the rational taufunctions. The vanishing properties of these taufunctions are investigated.
2000 Webs, Lenard schemes, and the local geometry of bihamiltonian Toda and Lax structures
 Selecta Math. (N.S
"... Abstract. We introduce a criterion that a given bihamiltonian structure allows a local coordinate system where both brackets have constant coefficients. This criterion is applied to the bihamiltonian open Toda lattice in a generic point, which is shown to be locally isomorphic to a Kronecker odddim ..."
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Abstract. We introduce a criterion that a given bihamiltonian structure allows a local coordinate system where both brackets have constant coefficients. This criterion is applied to the bihamiltonian open Toda lattice in a generic point, which is shown to be locally isomorphic to a Kronecker odddimensional pair of brackets with constant coefficients. This shows that the open Toda lattice cannot be locally represented as a product of two bihamiltonian structures. In a generic point the bihamiltonian periodic Toda lattice is shown to be isomorphic to a product of two open Toda lattices (one of which is a (trivial) structure of dimension 1). While the above results might be obtained by more traditional methods, we use an approach based on general results on geometry of webs. This demonstrates a possibility to apply a geometric language to problems on bihamiltonian integrable systems, such a possibility may be no less important than the particular results proven in this paper. Based on these geometric approaches, we conjecture that decompositions similar
Master symmetries and biHamiltonian structures for the relativistic Toda lattice
, 1997
"... We define a biHamiltonian formulation for the relativistic Toda lattice with a recursion operator on IR 2n . We use a theorem of W. Oevel to generate higher order Poisson tensors and master symmetries for the relativistic Toda lattice. These Poisson tensors and master symmetries reduce to IR 2n\ ..."
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We define a biHamiltonian formulation for the relativistic Toda lattice with a recursion operator on IR 2n . We use a theorem of W. Oevel to generate higher order Poisson tensors and master symmetries for the relativistic Toda lattice. These Poisson tensors and master symmetries reduce to IR 2n\Gamma1 . M. S. Classification (1991): 58F05. Key words: Relativistic Toda lattice, biHamiltonian system, master symmetry. 2 Introduction The relativistic Toda lattice (RTL) was introduced by S. N. Ruijsenaars [1] and has been studied by many authors, in particular M. Bruschi and O. Ragnisco [2, 3], W. Oevel et al. [4], Yu. B. Suris [5] and P. Damianou [6]. It is a finitedimensional completely integrable biHamiltonian system. Its biHamiltonian formulation and its complete integrability were proven by using various methods: Lax representation [3], [6], master symmetries [4], [6], recursion operators [2], [4]. Master symmetries were introduced by A. S. Fokas and B. Fuchssteiner [9] and...