Том 242, № 5 (2019)

We consider the problem of coordinatizing the manifold constructed via the Marsden–Weinstein quotient. Rational canonical coordinates on the symplectic reduction with respect to the diagonal action of the general linear group on the Cartesian product of coadjoint orbits in the case of the complex general linear group are constructed on an algebraically open subset of the quotient space. The method is based on an iterative process of constructing projection-flag coordinates, and works if the matrices constituting the orbits have a sufficiently rich set of invariant subspaces.

In this paper, we show that an operator introduced by A. A. Tsvetkov enjoys all the necessary properties of a Q-operator. It is shown that the Q-operator of the XXX spin chain with spin ℓ turns into Tsvetkov’s operator in the continuous limit as ℓ→∞. Bibliography: 18 titles.

The exactly solvable four-vertex model on a square grid with the fixed boundary conditions in a presence of a special external field is considered. Namely, we study a system in a linear field acting on the central column of the grid. The partition function of the model is calculated by the quantum inverse scattering method. The answer is written in determinantal form.

We obtain differential operators for the bivariate Chebyshev polynomials of the first kind associated with the root systems of the simple Lie algebras C₂ and G₂.

Earlier, in the study of combinatorial properties of the heat kernel of the Laplace operator with covariant derivative, a diagram technique and matrix formalism were constructed. In particular, the obtained formalism allows one to control the coefficients of the heat kernel, which is useful for calculations. In this paper, we consider a simple case with an Abelian connection in the two-dimensional space. This model allows us to give a mathematical description of the operators and find a relation between these operators and generating functions of numbers.

In this note, we derive the asymptotical behavior of local correlation functions in dimer models on a domain of the hexagonal lattice in the continuum limit, when the size of the domain goes to infinity and the parameters of the model scale appropriately.

In this paper, we investigate genus 2 algebro-geometric solutions of the AKNS hierarchy equations strictly periodic with respect to the space variable x. In general, genus 2 solutions, which are expressed in terms of two-dimensional Riemann theta functions, are not strictly periodic in x. We show that x-periodic solutions can be obtained by an appropriate choice of a hyperelliptic spectral curve having a structure of a covering of an elliptic curve. For odd-numbered members of the AKNS hierarchy, these solutions can be made periodic also with respect to the corresponding time variables of the AKNS hierarchy, by imposing further restrictions on the structure of the spectral curve. The corresponding solutions are especially interesting from the point of view of potential applications to the study of signal propagation in nonlinear optical fibers.