Unitary Representations of the Quantum Lorentz Group and Quantum Relativistic Toda Chain

We give a group theory interpretation of the three types of q-Bessel functions. We consider a family of quantum Lorentz groups and a family of quantum Lobachevsky spaces. For three values of the parameter of the quantum Lobachevsky space, the Casimir operators correspond to the two-body relativistic open Toda-chain Hamiltonians whose eigenfunctions are the modified q-Bessel functions of the three types. We construct the principal series of unitary irreducible representations of the quantum Lorentz groups. Special matrix elements in the irreducible spaces given by the q-Macdonald functions are the wave functions of the two-body relativistic open Toda chain. We obtain integral representations for these functions.     

Electron zitterbewegung in a linear crystal with alternating parity

It is shown thatin a linear periodic chain at whose sites electron wave functions of opposite parities alternate the existence of zitterbewegung of a nonrelativistic electron can be established without passage to the limiting case of a continuum. An analogy is established with the relations obtained earlier by the authors for the motion of a nonrelativistic electron in a two-site system. The zitterbewegung of the nonrelativistic electron in the considered system is given an interpretation analogous to the interpretation obtained by the first two authors for the zitterbewegung of a relativistic electron in Dirac's theory. A connection between the energy and effective mass of the nonrelativistic electron in the considered system is established.Institute of Metal Physics, Urals Branch of the Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 2, pp. 343–352, February, 1993.     

The exponential function rational expansion method and exact solutions to nonlinear lattice equations system

We propose an exponential function rational expansion method for solving exact traveling wave solutions to nonlinear differential-difference equations system. By this method, we obtain some exact traveling wave solutions to the relativistic Toda lattice equations system and discuss the significance of these solutions. Finally, we give an open problem.     

Biorthogonal Laurent Polynomials,Toplitz Determinants,Minimal Toda Orbits and Isomonodromic Tau Functions

We consider the class of biorthogonal polynomials that are used to solve the inverse spectral problem associated to elementary co-adjoint orbits of the Borel group of upper triangular matrices; these orbits are the phase space of generalized integrable lattices of Toda type. Such polynomials naturally interpolate between the theory of orthogonal polynomials on the line and orthogonal polynomials on the unit circle and tie together the theory of Toda, relativistic Toda, Ablowitz-Ladik and Volterra lattices. We establish corresponding Christoffel-Darboux formulae. For all these classes of polynomials a 2 × 2 system of Differential-Difference-Deformation equations is analyzed in the most general setting of pseudo-measures with arbitrary rational logarithmic derivative. They provide particular classes of isomonodromic deformations of rational connections on the Riemann sphere. The corresponding isomonodromic tau function is explicitly related to the shifted Toplitz determinants of the moments of the pseudo-measure. In particular, the results imply that any (shifted) Toplitz (Hankel) determinant of a symbol (measure) with arbitrary rational logarithmic derivative is an isomonodromic tau function.     

A hierarchy of differential-difference equations,conservation laws and new integrable coupling system

Starting from a discrete spectral problem with two arbitrary parameters, a hierarchy of nonlinear differential-difference equations is derived. The new hierarchy not only includes the original hierarchy, but also the well-known Toda equation and relativistic Toda equation. Moreover, infinitely many conservation laws for a representative discrete equation are given. Further, a new integrable coupling system of the resulting hierarchy is constructed.     

Construction of a new class of quantum integrable inhomogeneous models

Integrable inhomogenous or impurity models are usually constructed by either shifting the spectral parameter in the Lax operator or using another representation of the spin algebra. We propose a more involved general method for such construction in which the Lax operator contains generators of a novel quadratic algebra, a generalization of the known quantum algebra. In forming the monodromy matrix, we can replace any number of the local Lax operators with different realizations of the underlying algebra, which can result in spin chains with nonspin impurities causing changed coupling across the impurity sites, as well as with impurities in the form of bosonic operators. Following the same idea, we can also generate integrable inhomogeneous versions of the generalized lattice sine-Gordon model, nonlinear Schrödinger equation, Liouville model, relativistic and nonrelativistic Toda chains, etc.     

Webs, Lenard schemes, and the local geometry of bi-Hamiltonian Toda and Lax structures

We introduce a criterion that a given bi-Hamiltonian structure admits a local coordinate system where both brackets have constant coefficients. This criterion is applied to the bi-Hamiltonian open Toda lattice in a generic point, which is shown to be locally isomorphic to a Kronecker odd-dimensional pair of brackets with constant coefficients. This shows that the open Toda lattice cannot be locally represented as a product of two bi-Hamiltonian structures. Near, a generic point, the bi-Hamiltonian 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 the possibility of applying a geometric language to problems on bi-Hamiltonian integrable systems; such a possibility may be no less important than the particular results proved in this paper. Based on these geometric approaches, we conjecture that decompositions similar to the decomposition of the periodic Toda lattice exist in local geometry of the Volterra system, the complete Toda lattice, the multidimensional Euler top, and a regular bi-Hamiltonian Lie coalgebra. We also state general conjectures about the geometry of more general "homogeneous" finite-dimensional bi-Hamiltonian structures. The class of homogeneous structures is shown to coincide with the class of systems integrable by Lenard scheme. The bi-Hamiltonian structures which admit a non-degenerate Lax structure are shown to be locally isomorphic to the open Toda lattice.     

Exactly solvable models in supersymmetric quantum mechanics and connection with spectrum-generating algebras

Analytic expressions for the eigenvalues and eigenfunctions of nonrelativistic shape-invariant Hamiltonians can be derived using the well-known method of supersymmetric quantum mechanics. Most of these Hamiltonians also possess spectrum-generating algebras and are hence solvable by an independent group theory method. We demonstrate the equivalence of the two solution methods by developing an algebraic framework for shape-invariant Hamiltonians with a general parameter change involving nonlinear extensions of Lie algebras. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 3, pp. 362–374, March, 1999.     

A Kind of Discretization of the KdV Hierarchy

We propose a method for introducing higher-order terms in the potential expansion in order to study the continuum limits of the Toda hierarchy. These higher-order terms are differential polynomials in the lower-order terms. This type of potential expansion allows using fewer equations in the Toda hierarchy to recover the KdV hierarchy by the so-called recombination method. We show that the Lax pairs, the Poisson tensors, and the Hamiltonians of the Toda hierarchy tend toward the corresponding objects of the KdV hierarchy in the continuum limit. This method gives a kind of discretization of the KdV hierarchy.     

Matrix Model and Stationary Problem in the Toda Chain

We analyze the stationary problem for the Toda chain and show that the arising geometric data exactly correspond to the multisupport solutions of the one-matrix model with a polynomial potential. We calculate the Hamiltonians and symplectic forms for the first nontrivial examples explicitly and perform the consistency checks. We formulate the corresponding quantum problem and discuss some of its properties and prospects. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 3–16, January, 2006.     

Extended Relativistic Toda Lattice,L-Orthogonal Polynomials and Associated Lax Pair

When a measure \(\varPsi(x)\) on the real line is subjected to the modification \(d\varPsi^{(t)}(x) = e^{-tx} d \varPsi(x)\), then the coefficients of the recurrence relation of the orthogonal polynomials in \(x\) with respect to the measure \(\varPsi^{(t)}(x)\) are known to satisfy the so-called Toda lattice formulas as functions of \(t\). In this paper we consider a modification of the form \(e^{-t(\mathfrak{p}x+ \mathfrak{q}/x)}\) of measures or, more generally, of moment functionals, associated with orthogonal L-polynomials and show that the coefficients of the recurrence relation of these L-orthogonal polynomials satisfy what we call an extended relativistic Toda lattice. Most importantly, we also establish the so called Lax pair representation associated with this extended relativistic Toda lattice. These results also cover the (ordinary) relativistic Toda lattice formulations considered in the literature by assuming either \(\mathfrak{p}=0\) or \(\mathfrak{q}=0\). However, as far as Lax pair representation is concern, no complete Lax pair representations were established before for the respective relativistic Toda lattice formulations. Some explicit examples of extended relativistic Toda lattice and Langmuir lattice are also presented. As further results, the lattice formulas that follow from the three term recurrence relations associated with kernel polynomials on the unit circle are also established.

     

Hirota’s difference equations

A review of selected topics for Hirota’s bilinear difference equation (HBDE) is given. This famous three-dimensional difference equation is known to provide a canonical integrable discretization for most of the important types of soliton equations. Similar to continuous theory, HBDE is a member of an infinite hierarchy. The central point of our paper is a discrete version of the zero curvature condition explicitly written in the form of discrete Zakharov-Shabat equations for M-operators realized as difference or pseudo-difference operators. A unified approach to various types of M-operators and zero curvature representations is suggested. Different reductions of HBDE to two-dimensional equations are considered, with discrete counterparts of the KdV, sine-Gordon, Toda chain, relativistic Toda chain, and other examples. The article was written by the request of the Editorial Board. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 2, pp. 179–230, November, 1997.     

Morse homology for the heat flow

We use the heat flow on the loop space of a closed Riemannian manifold—viewed as a parabolic boundary value problem for infinite cylinders—to construct an algebraic chain complex. The chain groups are generated by perturbed closed geodesics. The boundary operator is defined by counting, modulo time shift, heat flow trajectories between geodesics of Morse index difference one. By Salamon and Weber (GAFA 16:1050–138, 2006) this heat flow homology is naturally isomorphic to Floer homology of the cotangent bundle for Hamiltonians given by kinetic plus potential energy.     

Integrability conditions for an analogue of the relativistic Toda chain

We consider a class of discrete-differential equations that contains the relativistic Toda chain and is characterized by one arbitrary function of six variables. We derive three conditions that allow testing the integrability of any given equation in this class. In deriving these conditions, we use higher symmetries distinguishing the equations that are integrable via the inverse scattering method. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 66–80, April, 2007.     

Wave functions of the Toda chain with boundary interaction

We give an integral representation of the wave functions of the quantum N-particle Toda chain with boundary interaction. In the case of the Toda chain with a one-boundary interaction, we obtain the wave function by an integral transformation from the wave functions of the open Toda chain. The kernel of this transformation is given explicitly in terms of -functions. The wave function of the Toda chain with a two-boundary interaction is obtained from the previous wave functions by an integral transformation. In this case, the difference equation for the kernel of the integral transformation admits a separation of variables. The separated difference equations coincide with the Baxter equation.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 346–364, February, 2005.     

Approximate solution for system of differential-difference equations by means of the homotopy analysis method

In this paper, the homotopy analysis method is successfully applied to solve approximate solutions of the differential-difference system. The solution of another relativistic Toda lattice system is considered. Comparisons made between the results of the proposed method and exact solutions reveal that the homotopy analysis method is very effective in solving differential-difference system.     

The Discrete Relativistic Toda Molecule Equation and a Padé Approximation Algorithm

The relativistic Toda molecule equation (RTM) describes a one-parameter deformation of coefficients of the recurrence relation of a class of biorthogonal polynomials including the Szegö polynomials. In this paper, we present (i) explicit solutions of the discrete relativistic Toda molecule equation (d-RTM), (ii) a new Padé approximation algorithm for a given power series.     

Transition function for the Toda chain

We use the method of Λ-operators developed by Derkachov, Korchemsky, and Manashov to derive eigenfunctions for the open Toda chain. Using the diagram technique developed for these Λ-operators, we reproduce the Sklyanin measure and study the properties of the Λ-operators. This approach to the open Toda chain eigenfunctions reproduces the Gauss-Givental representation for these eigenfunctions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 3, pp. 371–390, March, 2007.     

Self-Dual Hamiltonians as Deformations of Free Systems

We formulate the problem of finding self-dual Hamiltonians (associated with integrable systems) as deformations of free systems given on various symplectic manifolds and discuss several known explicit examples including the recently found double elliptic Hamiltonians. We consider self-duality as the basic principle, while duality in integrable systems (of the Toda/Calogero/Ruijsenaars type) comes as a secondary notion (degenerations of self-dual systems).     

Deformations of Calogero-Moser systems and finite Toda chains

Recent results pertaining to the complete integrability of some noveln-particle models in dimension one are presented. These models generalize the Calogero-Moser systems related to classical root systems. Generalizations of the relativistic Toda chain are obtained via limit transitions.Department of Mathematics and Computer Science, University of Amsterdam, Platage Muidergracht 24, 1018 TV Amsterdam, The Netherlands. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 2, pp. 234–240, May, 1994.