integration of quantum many body problems by affine knizhnik zamolodchikov equations abstract an equivalence of the affine analogues of knizhnik zamolodchikov and dunkl equations is established for arbitrary root systems as a corollary an isomorphism is obtained from the solutions of the above kz equations in the induced representations of hecke algebras to the eigenfunctions for the corresponding matrix quantum calogero sutherland operators which generalizes the matsuo map

commuting difference operators with elliptic coefficients from baxter s vacuum vectors for quantum integrable models with elliptic r matrix we construct the baxter q operator in infinite dimensional representations of the algebra of observables

backlund transformations as exact integrable time discretizations for the trigonometric gaudin model we construct a two parameter family of backlund transformations for the trigonometric classical gaudin magnet the approach follows closely the one introduced by sklyanin and kuznetsov 1998 j phys a math gen 31 2241 51 in a number of seminal papers and takes advantage of the intimate relation between the trigonometric and the rational case as in the paper by hone kuznetsov and one of the authors or 2001 j phys a math gen 34 2477 90 the backlund transformations are presented as explicit symplectic maps starting from their lax representation the expected connection with the xxz heisenberg chain is established and the rational xxx case is recovered in a suitable limit it is shown how to obtain a physical transformation mapping real variables into real variables the interpolating hamiltonian flow is derived and some numerical iterations of the map are presented

the index of the scattering operator on the positive spectral subspace we construct the scattering operator for a spinor field in a time dependent background by the dyson expansion then we show that the restriction of the scattering operator to the positive spectral subspace with respect to a reference hamiltonian is fredholm the computation of the index of this restriction is reduced to the index computation for an elliptic pseudodifferential operator of order zero we obtain the index in terms of a cohomological formula by means of the atiyah singer index theorem

relative index theorems and supersymmetric scattering theory we discuss supersymmetric scattering theory and employ krein s theory of spectral shift functions to investigate supersymmetric scattering systems this is the basis for the derivation of relative index theorems on some classes of open manifolds as an example we discuss the de rham complex for obstacles in n and asymptotically flat manifolds it is shown that the absolute or relative euler characteristic of an obstacle in n may be obtained from scattering data for the laplace operator on forms with absolute or relative boundary conditions respectively in the case of asymptotically flat manifolds we obtain the chern gauss bonnet theorem for thel2 euler characteristic

backlund transformations and hamiltonian flows in this work we show that under certain conditions parametric backlund transformations for a finite dimensional integrable system can be interpreted as solutions to the equations of motion defined by an associated non autonomous hamiltonian the two systems share the same constants of motion this observation leads to the identification of the hamiltonian interpolating the iteration of the discrete map defined by the transformations which indeed in numerical applications can be considered a linear combination of the integrals appearing in the spectral curve of the lax matrix an example with the periodic toda lattice is given

integrable symplectic maps abstract in this paper we first give a terse survey of symplectic maps their canonical formulation and integrability then we introduce a rigorous procedure to construct integrable symplectic maps starting from integrable evolution equations on lattices a number of illustrative examples are provided

peakons r matrix and toda lattice abstract the integrability of a family of hamiltonian systems describing in a particular case the motion of n peakons special solutions of the so called camassa holm equation is established in the framework of the r matrix approach starting from its lax representation in the general case the r matrix is a dynamical one and has an interesting though complicated structure however for a particular choice of the relevant parameters in the hamiltonian the one corresponding to the pure peakons case the r matrix becomes essentially constant and reduces to the one pertaining to the finite non periodic toda lattice intriguing consequences of this property are discussed and an integrable time discretisation is derived

growth and integrability in the dynamics of mappings the growth of some numerical characteristics of the mappings under their iterations in the context of the general problem of integrability is discussed in the general case such characteristics as complexity by arnold or the number of the different images for the multiple valued mappings are growing exponentially it is shown that the integrability is closely related with thepolynomial growth the analogies with quantum integrable systems are discussed

backlund transformations for many body systems related to kdv we present backlund transformations bts with parameter for certain classical integrable n body systems namely the many body generalized henon heiles garnier and neumann systems our construction makes use of the fact that all these systems may be obtained as particular reductions stationary or restricted flows of the kdv hierarchy alternatively they may be considered as examples of the reduced sl 2 gaudin magnet the bts provide exact time discretizations of the original continuous systems preserving the lax matrix and hence all integrals of motion and satisfy the spectrality property with respect to the backlund parameter

discrete pluriharmonic functions as solutions of linear pluri lagrangian systems pluri lagrangian systems are variational systems with the multi dimensional consistency property this notion has its roots in the theory of pluriharmonic functions in the z invariant models of statistical mechanics in the theory of variational symmetries going back to noether and in the theory of discrete integrable systems a d dimensional pluri lagrangian problem can be described as follows given a d form l on an m dimensional space m d whose coefficients depend on a function u of m independent variables called field find those fields u which deliver critical points to the action functionals s sigma int sigma l for any d dimensional manifold in the m dimensional space we investigate discrete 2 dimensional linear pluri lagrangian systems i e those with quadratic lagrangians l the action is a discrete analogue of the dirichlet energy and solutions are called discrete pluriharmonic functions we classify linear pluri lagrangian systems with lagrangians depending on diagonals they are described by generalizations of the star triangle map examples of more general quadratic lagrangians are also considered

the non dynamical r matrix structure for the elliptic calogero moser model in this paper we construct a new lax operator for the elliptic calogero moser model with general from the classical dynamical twisting in which the corresponding r matrix is purely numerical non dynamic the non dynamical r matrix structure of this lax operator is obtained which is the elliptic symmetric r matrix

the periodic toda chain and a matrix generalization of the bessel function recursion relations the authors obtain the quantization conditions of the periodic toda lattice in the baxter form lambda u q u inq u i i nq u i lambda is the transfer matrix containing the information about the spectrum and q is an integral operator commuting with lambda the logarithms of the matrix elements of q are the generating functions of the canonical backlund transformation the requirement that q is analytic and vanishes when u goes to infinity completely determines the spectrum of lambda

source identity and kernel functions for inozemtsev type systems the inozemtsev hamiltonian is an elliptic generalization of the differential operator defining the bcn trigonometric quantum calogero sutherland model and its eigenvalue equation is a natural many variable generalization of the heun differential equation we present kernel functions for inozemtsev hamiltonians and chalykh feigin veselov sergeev type deformations thereof our main result is a solution of a heat type equation for a generalized inozemtsev hamiltonian which is the source of all these kernel functions applications are given including a derivation of simple exact eigenfunctions and eigenvalues of the inozemtsev hamiltonian

baxter s q operator for the homogeneous xxx spin chain applying the pasquier gaudin procedure we construct baxter s q operator for the homogeneous xxx model as an integral operator in the standard representation of sl 2 the connection between the q operator and the local hamiltonians is discussed we show that lipatov s duality symmetry operator arises naturally as the leading term of the asymptotic expansion of the q operator for large values of the spectral parameter

multi time lagrangian 1 forms for families of backlund transformations toda type systems general lagrangian theory of discrete one dimensional integrable systems is illustrated by a detailed study of backlund transformations for toda type systems commutativity of backlund transformations is shown to be equivalent to the consistency of the system of discrete multi time euler lagrange equations the precise meaning of the commutativity in the periodic case when all maps are double valued is established it is shown that the gluing of different branches is governed by the so called superposition formulas the closure relation for the multi time lagrangian 1 form on solutions of the variational equations is proved for all toda type systems superposition formulas are instrumental for this proof the closure relation was previously shown to be equivalent to the spectrality property of backlund transformations i e to the fact that the derivative of the lagrangian with respect to the spectral parameter is a common integral of motion of the family of backlund transformations we relate this integral of motion to the monodromy matrix of the zero curvature representation which is derived directly from equations of motion in an algorithmic way this serves as further evidence in favor of the idea that backlund transformations serve as zero curvature representations for themselves

a q difference baxter operator for the ablowitz ladik chain we construct the baxter operator and the corresponding baxter equation for a quantum version of the ablowitz ladik model the result is achieved in two different ways by using the well known bethe ansatz technique and by looking at the quantum analogue of the classical backlund transformations general results about integrable models governed by the same r matrix algebra will be given baxter s equation comes out to be a q difference equation involving both the trace and the quantum determinant of the monodromy matrix the spectrality property of the classical backlund transformations gives a trace formula representing the classical analogue of baxter s equation a q integral representation of the baxter operator is discussed

algebraic entropy for algebraic maps we propose an extension of the concept of algebraic entropy as introduced by bellon and viallet for rational maps to algebraic maps or correspondences of a certain kind the corresponding entropy is an index of the complexity of the map the definition inherits the basic properties from the definition of entropy for rational maps we give an example with positive entropy as well as two examples taken from the theory of backlund transformations

the massive feynman propagator on asymptotically minkowski spacetimes we consider the massive klein gordon equation on asymptoti cally minkowski spacetimes in the sense that the manifold is r 1 d and the metric approaches that of minkowski space at infinity in a short range way jointly in time and space variables in this setup we define feynman and anti feynman scattering data and prove the fredholm property of the klein gordon operator with the associated atiyah patodi singer type boundary conditions at infinite times we then construct a parametrix with compact remainder terms for the fredholm problem and prove that it is also a feynman parametrix in the sense of duistermaat and hormander

kernel functions and backlund 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 backlund 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

separation of variables for the ruijsenaars model and a new integral representation for the macdonald polynomials using the baker akhiezer function technique we construct a separation of variables for the classical trigonometric three particle ruijsenaars model a relativistic generalization of the calogero moser sutherland model in the quantum case an integral operator m is constructed from the askey wilson contour integral the operator m transforms the eigenfunctions of the commuting hamiltonians the macdonald polynomials for the root sytem into the factorized form where s y is a laurent polynomial of one variable expressed in terms of the basic hypergeometric series the inversion of m produces a new integral representation for the macdonald polynomials we also present some results and conjectures for the general n particle case

canonical explicit backlund transformations with spectrality for constrained flows of soliton hierarchies abstract it is shown that explicit backlund transformations bts for the high order constrained flows of soliton hierarchy can be constructed via their darboux transformations and lax representation and these bts are canonical transformations including backlund parameter and possess a spectrality property with respect to and the conjugated variable for which the pair lies on the spectral curve as model we present the canonical explicit bts with the spectrality for high order constrained flows of the kaup newell hierarchy and the kdv hierarchy

an index theorem for lorentzian manifolds with compact spacelike cauchy boundary we show that the dirac operator on a compact globally hyperbolic lorentzian spacetime with spacelike cauchy boundary is a fredholm operator if appropriate boundary conditions are imposed we prove that the index of this operator is given by the same expression as in the index formula of atiyah patodi singer for riemannian manifolds with boundary the index is also shown to equal that of a certain operator constructed from the evolution operator and a spectral projection on the boundary in case the metric is of product type near the boundary a feynman parametrix is constructed

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

a q operator for the twisted xxx model taking the isotropic limit 1 in a recent representation theoretic construction of baxter s q operators for the xxz model with quasi periodic boundary conditions we obtain new results for the xxx model we show that quasi periodic boundary conditions are needed to ensure convergence of the q operator construction and derive a quantum wronskian relation which implies two different sets of bethe ansatz equations one above the other below the equator of total spin sz 0 we discuss the limit to periodic boundary conditions at the end and explain how this construction relates to the trace functional introduced by boos et al in the context of correlation functions on the infinite lattice we also identify a special subclass of solutions to the quantum wronskian and numerically verify them up to spin chains of ten sites this special type of solutions might persist for longer chains

new hierarchies of isospectral and non isospectral integrable nlees derived from the harry dym spectral problem abstract by making use of our previous framework we display the hierarchies of generalized nonlinear evolution equations gnlees associated with the harry dym spectral problem and construct the corresponding generalized lax representations glr in this paper it will be clearly seen that the generalized hierarchies can include not only the well known harry dym hierarchy of isospectral nlees but also other new integrable hierarchies of isospectral and non isospectral nlees through choosing the so called seed function these new hierarchies of nlees give some new integrable evolution equations which are very likely to have potential applications in theoretical and experimental physics all of these hierarchies of nlees possess the glr

generalized bethe ansatz with the general spin representations of the sklyanin algebra the quantum inverse scattering method is applied to the inhomogeneous heisenberg xyz model with general spins i e the general spin representations of the sklyanin algebra by means of the generalized bethe ansatz developed by takhtajan and faddeev 1979 the eigenvectors and eigenvalues of the transfer matrix are found

the index of scattering operators of dirac equations ii abstract the index formula of scattering operators of the previous paper of the author t matsui the index of scattering operators of dirac equations commun math phys 110 1987 553 571 is shown to hold for a wider class of potentials which contains both gauge potentials with compactly supported energy and instantontype potentials

r matrix quantization of the elliptic ruijsenaars schneider model it is shown that the classical l operator algebra of the elliptic ruijsenaars schneider model can be realized as a subalgebra of the algebra of functions on the cotangent bundle over the centrally extended current group in two dimensions it is governed by two dynamic and bar r matrices satisfying a closed system of equations the corresponding quantum r and bar r matrices are found as solutions to quantum analogues of these equations we present the quantum l operator algebra and show that the system of equations for r and bar r arises as the compatibility condition for this algebra it turns out that the r matrix is twist equivalent to the felder elliptic rf matrix with bar r playing the role of the twist the simplest representation of the quantum l operator algebra corresponding to the elliptic ruijsenaars schneider model is obtained the connection of the quantum l operator algebra to the fundamental relation rb ll llrb with the belavin elliptic r matrix is established as a by product of our construction we find a new n parameter elliptic solution to the classical yang baxter equation

an integrable discretization of the rational mathfrak su 2 gaudin model and related systems the first part of the present paper is devoted to a systematic construction of continuous time finite dimensional integrable systems arising from the rational su 2 gaudin model through certain contraction procedures in the second part we derive an explicit integrable poisson map discretizing a particular hamiltonian flow of the rational su 2 gaudin model then the contraction procedures enable us to construct explicit integrable discretizations of the continuous systems derived in the first part of the paper

some integrable maps and their hirota bilinear forms we introduce a two parameter family of birational maps which reduces to a family previously found by demskoi tran van der kamp and quispel dtkq when one of the parameters is set to zero the study of the singularity confinement pattern for these maps leads to the introduction of a tau function satisfying a homogeneous recurrence which has the laurent property and the tropical or ultradiscrete analogue of this homogeneous recurrence confirms the quadratic degree growth found empirically by demskoi et al we prove that the tau function also satisfies two different bilinear equations each of which is a reduction of the hirota miwa equation also known as the discrete kp equation or the octahedron recurrence furthermore these bilinear equations are related to reductions of particular two dimensional integrable lattice equations of discrete kdv or discrete toda type these connections as well as the cluster algebra structure of the bilinear equations allow a direct construction of poisson brackets lax pairs and first integrals for the birational maps as a consequence of the latter results we show how each member of the family can be lifted to a system that is integrable in the liouville sense clarifying observations made previously in the original dtkq case

multi time lagrangian 1 forms for families of backlund transformations relativistic toda type systems we establish the pluri lagrangian structure for families of backlund transformations of relativistic toda type systems the key idea is a novel embedding of these discrete time one dimensional systems into certain two dimensional 2d pluri lagrangian lattice systems this embedding allows us to identify the corner equations which are the main building blocks of the multi time euler lagrange equations with local superposition formulae for backlund transformations these superposition formulae in turn are key ingredients necessary to understand and to prove commutativity of the multi valued backlund transformations furthermore we discover a 2d generalization of the spectrality property known for families of backlund transformations this result produces a family of local conservations laws for 2d pluri lagrangian lattice systems with densities being derivatives of the discrete 2 form with respect to the backlund spectral parameter thus a relation of the pluri lagrangian structure with more traditional integrability notions is established

backlund transformations for high order constrained flows of the akns hierarchy canonicity and spectrality property a new infinite number of one and two point backlund transformations bts with explicit expressions are constructed for the high order constrained flows of the akns hierarchy it is shown that these bts are canonical transformations including the backlund parameter and a spectrality property holds with respect to and the conjugated variable for which the point belongs to the spectral curve in addition the formulae of m times repeated darboux transformations for the high order constrained flows of the akns hierarchy are presented

l a b representation for nonlinear evolution equations and its applications in this paper we consider the l a b representation of nonlinear evolution equations nlees and present an approach to determine its range the method we adopt is that of lie quotient algebras as an example we consider the kaup newell spectral problem and give a category of nlees associated with it particularly we obtain the negative order hierarchies of nlees by considering the inverse of recursion operator

separation of variables and backlund transformations for the symmetric lagrange top we construct the one and two point integrable maps backlund transformations for the symmetric lagrange top we show that the lagrange top has the same algebraic poisson structure that belongs to the sl 2 gaudin magnet the two point map leads to a real time discretization of the continuous flow therefore it provides an integrable numerical scheme for integrating the physical flow we illustrate the construction by a few pictures of the discrete flow calculated in matlab

complete integrability of relativistic calogero moser systems and elliptic function identities poincare invariant generalizations of the galilei invariant calogero moser a particle systems are studied a quantization of the classical integrals s1 sn is presented such that the operators si9 sn mutually commute as a corollary it follows that si9 9sn poisson commute these results hinge on functional equations satisfied by the weierstrass and 0 functions a generalized cauchy identity involving the function leads to an n x n matrix l whose symmetric functions are proportional to sl5 9sn

on an integrable discretisation of the ablowitz ladik hierarchy following the general results on the relationships about backlund transformations bts and exact discretisation given in a previous work zullo f backlund transformations and hamiltonian flows j phys a math theor 46 145203 2013 10 1088 1751 8113 46 14 145203 we consider the ablowitz ladik hierarchy and a corresponding family of bts after discussing the boundary conditions we show how to get explicit transformations the hamiltonian properties of the maps and of the discrete flows are examined the conditions on the parameters of the map giving exact discretisations are discussed finally analytical and numerical examples are given

backlund transformations for the sl 2 gaudin magnet elementary one and two point backlund transformations are constructed for the generic case of the sl 2 gaudin magnet the spectrality property is used to construct these explicitly given poisson integrable maps which are time discretizations of the continuous flows with any hamiltonian from the spectral curve of the 2 2 lax matrix

properties of the canonical transformations of the time for the toda lattice and the henon heiles system for the toda lattice and the henon heiles system we consider properties of the canonical transformations of the extended phase space which preserve integrability at the special values of integrals of motion the integral trajectories separated variables and the action variables are invariant under change of the time on the other hand mapping of the time induces a shift of the generating function of the backlund transformation

separation of variables for the type periodic toda lattice we prove separation of variables for the most general type periodic toda lattice with a lax matrix it is achieved by finding proper normalization for the corresponding baker akhiezer function separation of variables for all other periodic toda lattices associated with infinite series of root systems follows by taking appropriate limits

the analogue of the backlund transformation for integrable many body systems canonical transformations analogous to the backlund transformation are discussed explicit formulae are given for many body systems of particles interacting in one dimension

quantum integrable systems and representations of lie algebras in this paper the quantum integrals of the hamiltonian of the quantum many body problem with the interaction potential k sinh2 x sutherland operator are constructed as images of higher casimirs of the lie algebra gl n under a certain homomorphism from the center of u gl n to the algebra of differential operators in n variables a similar construction applied to the affine gl n at the critical level k n defines a correspondence between higher sugawara operators and quantum integrals of the hamiltonian of the quantum many body problem with the potential equal to constant times the weierstrass function this allows one to give a new proof of the olshanetsky perelomov theorem stating that this hamiltonian defines a completely integrable quantum system a new expression is also given for eigenfunctions of the quantum integrals of the sutherland operator as traces of intertwining operators between certain representations of gl n

dynamical r matrix for the elliptic ruijsenaars schneider system the classical r matrix structure for the generic elliptic ruijsenaars schneider model is presented it makes manifest the integrability of this model as well as of its discrete time version that was constructed in a recent paper

configurations of points and topology of real line arrangements a central question in the study of line arrangements in the complex projective plane mathbb c mathbb p 2 is the following when does the combinatorial data of the arrangement determine its topological properties in the present work we introduce a topological invariant of complexified real line arrangements called the chamber weight this invariant is based on the weight counting over the points of the associated dual configuration located in particular chambers of the real projective plane mathbb r mathbb p 2 using this dual setting we construct several examples of complexified real line arrangements with the same combinatorial data and different embeddings in mathbb c mathbb p 2 i e zariski pairs which are distinguished by this invariant in particular we obtain new zariski pairs of 13 15 and 17 lines defined over mathbb q and containing only double and triple points for each one of our examples we derive some degenerations containing points of multiplicity 2 3 and 5 which are also zariski pairs we compute explicitly the moduli space of the combinatorics of one of these examples and prove that it has exactly two connected components we also obtain three geometric characterizations of these components the existence of two smooth conics one tangent to six lines and the other containing six triple points as well as the collinearity of three specific triple points

simultaneous separation for the kowalevski and goryachev chaplygin gyrostats in the special case of zero square integral the kowalevski gyrostat and goryachev chaplygin gyrostat share a simple separation of variables originated from the 4 4 lax matrix

a super integrable discretization of the calogero model a time discretization that preserves the super integrability of the calogero model is obtained by application of the integrable time discretization of the harmonic oscillator to the projection method for the calogero model with continuous time in particular the difference equations of motion which provide an explicit scheme for time integration are explicitly presented for the two body case numerical results exhibit that the scheme conserves all the 3 conserved quantities of the two body calogero model with a precision of the machine epsilon times the number of iterations

the lax representation for an integrable class of relativistic dynamical systems we exhibit the lax pair for the class of relativistic dynamical systems recently introduced by ruijsenaars and schneider whose equations of motion read whereb is an arbitrary constant and the weierstrass elliptic function

rational top and its classical r matrix we construct a rational integrable system the rational top on a co adjoint orbit of sl n lie group it is described by the lax operator with spectral parameter and classical non dynamical skew symmetric r matrix in the case of the orbit of minimal dimension the model is gauge equivalent to the rational calogero moser cm system to obtain the results we represent the lax operator of the cm model in two different factorized forms without spectral parameter related to the spinless case and another one with the spectral parameter the latter gives rise to the rational top while the first one is related to generalized cremmer gervais r matrices the gauge transformation relating the rational top and cm model provides the classical rational version of the irf vertex correspondence from the geometrical point of view it describes the modification of bundles over degenerated elliptic curve in view of the symplectic hecke correspondence the rational top is related to the rational spin cm model possible applications and generalizations of the suggested construction are discussed in particular the obtained r matrix defines a class of kzb equations

integrable structure of conformal field theory ii q operator and ddv equation this paper is a direct continuation of 1 where we began the study of the integrable structures in conformal field theory we show here how to construct the operators bf q pm lambda which act in the highest weight virasoro module and commute for different values of the parameter these operators appear to be the cft analogs of the q matrix of baxter 2 in particular they satisfy baxter s famous t q equation we also show that under natural assumptions about analytic properties of the operators as the functions of the baxter s relation allows one to derive the nonlinear integral equations of destri de vega ddv 3 for the eigenvalues of the q operators we then use the ddv equation to obtain the asymptotic expansions of the q operators at large it is remarkable that unlike the expansions of the t operators of 1 the asymptotic series for q contains the dual nonlocal integrals of motion along with the local ones we also discuss an intriguing relation between the vacuum eigenvalues of the q operators and the stationary transport properties in the boundary sine gordon model on this basis we propose a number of new exact results about finite voltage charge transport through the point contact in the quantum hall system

a quantum algebra approach to basic multivariable special functions generalizations to a base q of the appell and lauricella hypergeometric functions are studied within the framework of the representation theory of quantum algebras of type an this allows deriving various identities and formulae involving these q functions

on backlund transformations for many body systems using the n particle periodic toda lattice and the relativistic generalization due to ruijsenaars of the elliptic calogero moser system as examples we revise the basic properties of the backlund transformations bts from the hamiltonian point of view the analogy between the bt and baxter s quantum q operator pointed out by pasquier and gaudin is exploited to produce a conjugated variable for the parameter of the bt such that belongs to the spectrum of the lax operator as a consequence the generating function of the composition of n bts gives rise to another canonical transformation separating variables for the model for the toda lattice the dual bt parametrized by is introduced

discrete time toda systems in this paper we discuss several concepts of the modern theory of discrete integrable systems including time discretization based on the notion of backlund transformation symplectic realizations of multi hamiltonian structures interrelations between discrete 1d systems and lattice 2d systems multi dimensional consistency as integrability of discrete systems interrelations between integrable systems of quad equations and integrable systems of laplace type pluri lagrangian structure as integrability of discrete variational systems all these concepts are illustrated by the discrete time toda lattices and their relativistic analogs

quantum backlund transformation for the integrable dst model for the integrable case of the discrete self trapping dst model we construct a backlund transformation the dual lax matrix and the corresponding dual backlund transformation are also found and studied the quantum analogue of the backlund transformation q operator is constructed as the trace of a monodromy matrix with an infinite dimensional auxiliary space we present the q operator as an explicit integral operator as well as describing its action on the monomial basis as a result we obtain a family of integral equations for multivariable polynomial eigenfunctions of the quantum integrable dst model these eigenfunctions are special functions of the heun class which is beyond the hypergeometric class the integral equations found are new and they shall provide a basis for efficient analytical and numerical studies of such complicated functions

separation of variables for the ruijsenaars system we construct a separation of variables for the classical n particle ruijsenaars system the relativistic analog of the elliptic calogero moser system the separated coordinates appear as the poles of the properly normalised eigenvector baker akhiezer function of the corresponding lax matrix two different normalisations of the ba functions are analysed the canonicity of the separated variables is verified with the use of the r matrix technique the explicit expressions for the generating function of the separating canonical transform are given in the simplest cases n 2 and n 3 taking the nonrelativistic limit we also construct a separation of variables for the elliptic calogero moser system

integrable time discretisation of the ruijsenaars schneider model an exactly integrable symplectic correspondence is derived which in a continuum limit leads to the equations of motion of the relativistic generalization of the calogero moser system that was introduced for the first time by ruijsenaars and schneider for the discrete time model the equations of motion take the form of bethe ansatz equations for the inhomogeneous spin 1 2 xyz heisenberg magnet we present a lax pair the sympletic structure and prove the involutivity of the invariants exact solutions are investigated in the rational and hyperbolic trigonometric limits of the system that is given in terms of elliptic functions these solutions are connected with discrete soliton equations the results obtained allow us to consider the bethe ansatz equations as ones giving an integrable symplectic correspondence mixing the parameters of the quantum integrable system and the parameters of the corresponding bethe wavefunction

integrable discretizations of the bogoyavlensky lattices discretizations of the bogoyavlensky lattices are introduced belonging to the same hierarchies as the continuous time systems the construction exemplifies the general scheme for integrable discretization of systems on lie algebras with r matrix poisson brackets an initial value problem for the difference equations is solved in terms of a factorization problem in a group interpolating hamiltonian flows are found

backlund transformations for the elliptic gaudin model and a clebsch system a two parameters family of backlund transformations for the classical elliptic gaudin model is constructed the maps are explicit symplectic preserve the same integrals as for the continuous flows and are a time discretization of each of these flows the transformations can map real variables into real variables sending physical solutions of the equations of motion into physical solutions the starting point of the analysis is the integrability structure of the model it is shown how the analogue transformations for the rational and trigonometric gaudin model are a limiting case of this one an application to a particular case of the clebsch system is given

quantum integrable systems and elliptic solutions of classical discrete nonlinear equations in spite of the diversity of solvable models of quantum field theory and the vast variety of methods the final results display dramatic unification the spectrum of an integrable theory with a local interaction is given by a sum of elementary energies e sum limits i varepsilon left u i right 1 1 where u i obey a system of algebraic or transcendental equations known as bethe equations 1 2 the major ingredients of bethe equations are determined by the algebraic structure of the problem a typical example of a system of bethe equations related to a i type models with elliptic r matrix is e 4 eta nu frac phi left u j right phi left u j 2 right mathop pi limits k frac sigma left eta left u j u k 2 right right sigma left eta left u j u k 2 right right 1 2 where x is the weierstrass function and phi left u right mathop pi limits k 1 n sigma left eta left u y k right right 1 3

baxter t q equation for shape invariant potentials the finite gap potentials case the darboux transformation applied recurrently on a schrodinger operator generates what is called a dressing chain or from a different point of view a set of supersymmetric shape invariant potentials the finite gap potential theory is a special case of the chain for the finite gap case the equations of the chain can be expressed as a time evolution of a hamiltonian system we apply sklyanin s method of separation of variables to the chain we show that the classical equation of the separation of variables is the baxter t q relation after quantization

tetrahedra octahedra and cubo octahedra integrable geometry of multi ratios geometric and algebraic aspects of multi ratios m2n are investigated in detail connections with menelaus theorem clifford configurations and maxwell s reciprocal quadrangles are utilized to associate the multi ratios m4 m8 and m8 with tetrahedra octahedra and cubo octahedra respectively integrable maps defined on face centred fcc lattices and irregular lattices composed of the face centres of simple cubic lattices are constructed and related to the discrete kp and bkp equations and the integrable discrete darboux system governing conjugate lattices an interpretation in terms of integrable irregular lattices of slopes on the plane is also given