We study the perturbative power series expansions of the eigenvalues and eigenvectors of a general tridiagonal (Jacobi) matrix of dimension d . The (small) expansion parameters are the entries of the two diagonals of length d −1 sandwiching the principal diagonal that gives the unperturbed spectrum. The solution is found explicitly in terms of multivariable (Horn-type) hypergeometric series in 3 d −5 variables in the generic case. To derive the result, we first rewrite the spectral problem for the Jacobi matrix as an equivalent system of algebraic equations, which are then solved by the application of the multivariable Lagrange inversion formula. The corresponding Jacobi determinant is calculated explicitly. Explicit formulae are also found for any monomial composed of eigenvector's components.

We obtain a complete solution for the mean-field dynamics of the BCS paired state with a large, but finite number of Cooper pairs in the nonadiabatic regime. We show that the problem reduces to a classical integrable Hamiltonian system and derive a complete set of its integrals of motion. The condensate exhibits irregular multifrequency oscillations ergodically exploring the part of the phase space allowed by the conservation laws. In the thermodynamic limit, however, the system can asymptotically reach a steady state.

We study the mean-field dynamics of a fermionic condensate interacting with a single bosonic mode (a generalized Dicke model). This problem is integrable and can be mapped onto a corresponding BCS problem. We derive the general solution and a full set of integrals of motion for the time evolution of coupled Fermi-Bose condensates. The present paper complements our earlier study of the dynamics of the BCS model. Here we provide a self-contained introduction to the variable separation method, which enables a complete analytical description of the evolution of the generalized Dicke, BCS, and other similar models.

We develop an explicit description of a time-dependent response of fermionic condensates to perturbations.The dynamics of Cooper pairs at times shorter than the energy relaxation time can be described by the BCS model.We obtain a general explicit solution for the dynamics of the BCS model.We also solve a closely related dynamical problem -the central spin model, which describes a localized spin coupled to a "spin bath".Here, we focus on presenting the solution and describing its general properties, but also mention some applications, e.g. to nonstationary pairing in cold Fermi gases and to the issue of electron spin decoherence in quantum dots.A typical dynamics of the BCS and central spin models is quasi-periodic with a large number of frequencies and stable under small perturbations.We show that for certain special initial conditions the number of frequencies decreases and the solution simplifies.In particular, periodic solutions correspond to the ground state and excitations of the BCS model.

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.

Applying Baxter's method of the Q-operator to the set of Sekiguchi's commuting partial differential operators we show that Jack polynomials P(x_1,...,x_n) are eigenfunctions of a one-parameter family of integral operators Q_z. The operators Q_z are expressed in terms of the Dirichlet-Liouville n-dimensional beta integral. From a composition of n operators Q_{z_k} we construct an integral operator S_n factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator S_n admits a factorisation described in terms of restricted Jack polynomials P(x_1,...,x_k,1,...,1). Using the operator Q_z for z=0 we give a simple derivation of a previously known integral representation for Jack polynomials.

We present a geometric construction of Backlund transformations and discretizations for a large class of algebraic completely integrable systems. To be more precise, we construct families of Backlund transformations, which are naturally parametrized by the points on the spectral curve(s) of the system. The key idea is that a point on the curve determines, through the Abel-Jacobi map, a vector on its Jacobian which determines a translation on the corresponding level set of the integrals (the generic level set of an algebraic completely integrable systems has a group structure). Globalizing this construction we find (possibly multi-valued, as is very common for Backlund transformations) maps which preserve the integrals of the system, they map solutions to solutions and they are symplectic maps (or, more generally, Poisson maps). We show that these have the spectrality property, a property of Backlund transformations that was recently introduced. Moreover, we recover Backlund transformations and discretizations which have up to now been constructed by ad-hoc methods, and we find Backlund transformations and discretizations for other integrable systems. We also introduce another approach, using pairs of normalizations of eigenvectors of Lax operators and we explain how our two methods are related through the method of separation of variables.

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 x 4 Lax matrix.

The finite Pfaff lattice is given by commuting Lax pairs involving a finite matrix L (zero above the first subdiagonal) and a projection onto Sp(N). The lattice admits solutions such that the entries of the matrix L are rational in the time parameters t_1,t_2,..., after conjugation by a diagonal matrix. The sequence of polynomial tau-functions, solving the problem, belongs to an intriguing chain of subspaces of Schur polynomials, associated to Young diagrams, dual with respect to a finite chain of rectangles. Also, this sequence of tau-functions is given inductively by the action of a fixed vertex operator. As examples, one such sequence is given by Jack polynomials for rectangular Young diagrams, while another chain starts with any two-column Jack polynomial.

In 1889 Sophie Kowalevski published her famous paper `Sur le probleme de la rotation d'un corps solide autour d'un point fixe' (1889 Acta Math. 12 177-232). This paper earned her a prize from the Paris Academy of Sciences and fame as an outstanding mathematician. It has now become an important part of her mathematical legacy. Another well known and frequently cited result of her research is the Cauchy-Kowalevski theorem proved in her Doctoral Dissertation in 1874. In both works she approached the solution of differential equations in terms of analytic or meromorphic functions, and in both cases solution led to specific restrictions on the differential equations in question. In the case of the Cauchy-Kowalevski theorem she found a sufficient condition for a very general ODE to have an analytic solution. In the problem of the rotation of a solid body around a fixed point she invented a new method, subsequently known as the Kowalevski-Painlevé method, and found a new case for the parameters of the equations when there is an extra integral of motion. This meant that this special case, since then referred to as the Kowalevski top, can be integrated in quadratures. In modern terminology the Kowalevski top was one of the first examples of an integrable system. The actual integration of the equations of motion was a hard but challenging task, and required enormous intuition. Even today, any `simplification' of her ingenious solution is a challenging problem. These two of her major mathematical achievements were at the frontiers of mathematical research at the end of the 19th century and they opened the way to what now exists as the theory of integrable systems.

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 x 2 Lax matrix.

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 analog 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 describe 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 found integral equations are new and they shall provide a basis for efficient analytical and numerical studies of such complicated functions.

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.

We show that the method of separation of variables gives a natural generalization of integral relations for classical special functions of one variable. The approach is illustrated by giving a new proof of the “quadratic” integral relations for the continuous q-ultraspherical polynomials. The separating integral operator M expressed in terms of the Askey-Wilson operator is studied in detail: apart from writing down the characteristic (“separation”) equations it satisfies, we find its spectrum, eigenfunctions, inversion, invariants (invariant q-difference operators), and give its interpretation as a fractional q-integration operator. We also give expansions of the A1 Macdonald polynomials into the eigenfunctions of the separating operator M and vice versa.

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 Baecklund transformations (BT's) from the Hamiltonian point of view. The analogy between BT and Baxter's quantum Q-operator pointed out by Pasquier and Gaudin is exploited to produce a conjugated variable mu for the parameter lambda of the BT B_lambda such that mu belongs to the spectrum of the Lax operator L(lambda). As a consequence, the generating function of the composition of n BT's gives rise also to another canonical transformation separating variables for the model. For the Toda lattice the dual BT parametrized by mu is introduced.

: 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.

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.

The hidden symmetry of the quantum Calogero-Moser system with an inverse-square potential is algebraically demonstrated making use of Dunkl's operators. We find the underlying algebra explaining the super-integrability phenomenon for this system. Applications to related multi-variable Bessel functions are also discussed.

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.

q-Deformations of the Toda chains related to infinite series of Lie algebras and to their affine analogs are described. The corresponding 2×2 L-operators satisfying the standard quadratic algebra of XXZ-type are constructed. In the quantum case, the procedure of separation of variables is described. Bibliography: 14 titles.