The Poisson bracket compatible with the classical reflection equation algebra

We introduce a family of compatible Poisson brackets on the space of $2\times 2$ polynomial matrices, which contains the reflection equation algebra bracket. Then we use it to derive a multi-Hamiltonian structure for a set of integrable systems that includes the $XXX$ Heisenberg magnet with boundary conditions, the generalized Toda lattices and the Kowalevski top.Comment: 13 pages, LaTeX with AmsFont

Quadrics on complex Riemannian spaces of constant curvature, separation of variables, and the Gaudin magnet

Integrable systems that are connected with orthogonal separation of variables in complex Riemannian spaces of constant curvature are considered herein. An isomorphism with the hyperbolic Gaudin magnet, previously pointed out by one of the authors, extends to coordinates of this type. The complete classification of these separable coordinate systems is provided by means of the corresponding L matrices for the Gaudin magnet. The limiting procedures (or calculus) which relate various degenerate orthogonal coordinate systems play a crucial role in the classification of all such systems

Noncompact SL(2,R) spin chain

We consider the integrable spin chain model - the noncompact SL(2,R) spin magnet. The spin operators are realized as the generators of the unitary principal series representation of the SL(2,R) group. In an explicit form, we construct R-matrix, the Baxter Q-operator and the transition kernel to the representation of the Separated Variables (SoV). The expressions for the energy and quasimomentum of the eigenstates in terms of the Baxter Q-operator are derived. The analytic properties of the eigenvalues of the Baxter operator as a function of the spectral parameter are established. Applying the diagrammatic approach, we calculate Sklyanin's integration measure in the separated variables and obtain the solution to the spectral problem for the model in terms of the eigenvalues of the Q-operator. We show that the transition kernel to the SoV representation is factorized into a product of certain operators each depending on a single separated variable.Comment: 29 pages, 12 figure

Baxter Q-operator and Separation of Variables for the open SL(2,R) spin chain

We construct the Baxter Q-operator and the representation of the Separated Variables (SoV) for the homogeneous open SL(2,R) spin chain. Applying the diagrammatical approach, we calculate Sklyanin's integration measure in the separated variables and obtain the solution to the spectral problem for the model in terms of the eigenvalues of the Q-operator. We show that the transition kernel to the SoV representation is factorized into the product of certain operators each depending on a single separated variable. As a consequence, it has a universal pyramid-like form that has been already observed for various quantum integrable models such as periodic Toda chain, closed SL(2,R) and SL(2,C) spin chains.Comment: 29 pages, 9 figures, Latex styl

Linear rr-Matrix Algebra for Systems Separable\\ in Parabolic Coordinates

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 $2\times 2$ 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.Comment: 10 pages, LaTeX. Submitted to Phys.Lett.

An integrable discretization of the rational 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.Comment: 26 pages, 5 figure

Explicit solution of the (quantum) elliptic Calogero-Sutherland model

We derive explicit formulas for the eigenfunctions and eigenvalues of the elliptic Calogero-Sutherland model as infinite series, to all orders and for arbitrary particle numbers and coupling parameters. The eigenfunctions obtained provide an elliptic deformation of the Jack polynomials. We prove in certain special cases that these series have a finite radius of convergence in the nome $q$ of the elliptic functions, including the two particle (= Lam\'e) case for non-integer coupling parameters.Comment: v1: 17 pages. The solution is given as series in q but only to low order. v2: 30 pages. Results significantly extended. v3: 35 pages. Paper completely revised: the results of v1 and v2 are extended to all order

Separation of variables for the quantum SL(2,R) spin chain

We construct representation of the Separated Variables (SoV) for the quantum SL(2,R) Heisenberg closed spin chain and obtain the integral representation for the eigenfunctions of the model. We calculate explicitly the Sklyanin measure defining the scalar product in the SoV representation and demonstrate that the language of Feynman diagrams is extremely useful in establishing various properties of the model. The kernel of the unitary transformation to the SoV representation is described by the same "pyramid diagram" as appeared before in the SoV representation for the SL(2,C) spin magnet. We argue that this kernel is given by the product of the Baxter Q-operators projected onto a special reference state.Comment: 26 pages, Latex style, 9 figures. References corrected, minor stylistic changes, version to be publishe

Stochastic B\"acklund transformations

How does one introduce randomness into a classical dynamical system in order to produce something which is related to the `corresponding' quantum system? We consider this question from a probabilistic point of view, in the context of some integrable Hamiltonian systems

Electric field gradients in MgB2_2 synthesized at high pressure: 111^111Cd TDPAC study and ab initio calculation

We report the high-pressure synthesis of novel superconductor MgB$_2$ and some related compounds. The superconducting transition temperature of our samples of MgB$_2$ is equal to 36.6 K. The MgB$_2$ lattice parameters determined via X-ray diffraction are in excellent agreement with results of our ab initio calculations. The time-differential perturbed angular correlation (TDPAC) experiments demonstrate a small increase in quadrupole frequency of $^111$Cd probe with decreasing temperature from 293 to 4.2 K. The electric field gradient (EFG) at the B site calculated from first principles is in fair agreement with EFG obtained from $^11$B NMR spectra of MgB$_2$ reported in the literature. It is also very close to EFG found in our $^111$Cd TDPAC measurements, which suggests that the $^111$Cd probe substitutes for boron in the MgB$_2$ lattice.Comment: 10 pages, 3 figure