Vadim B. Kuznetsov

Linear r-matrix algebra for classical separable systems

We consider a hierarchy of the natural-type Hamiltonian systems of n degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates. We give a Lax representation in terms of 2*2 matrices for the whole hierarchy and construct the associated linear r-matrix algebra with the r-matrix dependent on the dynamical variables. A Yang-Baxter equation of dynamical type is proposed. Using the method of variable separation, we provide the integration of the systems in classical mechanics constructing the separation equations and, hence, the explicit form of action variables. The quantization problem is discussed with the help of the separation variables.

Quadrics on real Riemannian spaces of constant curvature: Separation of variables and connection with Gaudin magnet

The integrable systems are considered which are connected with separation of variables in real Riemannian spaces of constant curvature. An isomorphism is given for these systems with hyperbolic Gaudin magnet. Using this isomorphism, the complete classification of separable coordinate systems is provided by means of the corresponding L‐operators for the Gaudin magnet.

Quantum Bäcklund 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 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.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.

Bäcklund transformations for finite-dimensional integrable systems: a geometric approach

Abstract 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 parameterized 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.

SEPARATION OF VARIABLES FOR THE RUIJSENAARS SYSTEM

Abstract: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.

Linear r-matrix algebra for systems separable in parabolic coordinates

Abstract We consider a hierarchy of many particle systems on the line with polynomial potentials separable in parabolic coordinates. Using the Lax representation, written in terms of 2×2 matrices for the whole hierarchy, we construct the associated linear r -matrix algebra with the r -matrix dependent on the dynamical variables. A dynamical Yang-Baxter equation is discussed.Abstract We consider a hierarchy of many particle systems on the line with polynomial potentials separable in parabolic coordinates. Using the Lax representation, written in terms of 2×2 matrices for the whole hierarchy, we construct the associated linear r -matrix algebra with the r -matrix dependent on the dynamical variables. A dynamical Yang-Baxter equation is discussed.

Q-operator and factorised separation chain for Jack polynomials

Abstract Applying Baxters method of the Q -operator to the set of Sekiguchis commuting partial differential operators we show that Jack polynomials P λ (1/g) ( χ 1 , …, χ n ) …, χ 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 zk 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 λ (1/g) ( 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.

Equivalence of two graphical calculi

The author considers the integrable systems which are connected with separation of variables in the Helmholtz operator on the real Riemannian spaces of constant curvature. An isomorphism is given for these systems with a quantum hyperbolic Gaudin magnet. Using this isomorphism, the complete classification of all separable coordinate systems on the manifolds considered is provided by means of the corresponding L-operators for the Gaudin magnet.

HIDDEN SYMMETRY OF THE QUANTUM CALOGERO-MOSER SYSTEM

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

Nonequilibrium cooper pairing in the nonadiabatic regime

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.