For Belavin's elliptic quantum R-matrix, we construct an L-operator as a set of difference operators acting on functions on the type A weight space. According to the fundamental relation RLL = LLR, the trace of the L-operator gives a commuting difference operators. We show that for the above mentioned L-operator this approach gives Macdonald type operators with elliptic theta function coefficie...

In hep-th/0202087 it was argued that the operator L0 is bad defined in κ-basis as a kernel operator. Indeed, we show that L0 is a difference operator. We also find a representation of L1 and L−1 in a class of difference operators. On leave from Steklov Mathematical Institute, Moscow, Russia.

Let Ω be a convex domain with smooth boundary in Rd. It has been shown recently that the semigroup generated by the discrete Laplacian for quasi-uniform families of piecewise linear finite element spaces on Ω is analytic with respect to the maximum-norm, uniformly in the mesh-width. This implies a resolvent estimate of standard form in the maximum-norm outside some sector in the right halfplane...

Borg-type uniqueness theorems for matrix-valued Jacobi operators H and supersymmetric Dirac difference operators D are proved. More precisely, assuming reflectionless matrix coefficients A,B in the self-adjoint Jacobi operator H = AS + AS + B (with S the right/left shift operators on the lattice Z) and the spectrum of H to be a compact interval [E−, E+], E− < E+, we prove that A and B are certa...

Our aim is to set the foundations of a discrete vectorial calculus on uniform n-dimensional grids, that can be easily re-formulated on general irregular grids. As the key tool we first introduce the notion of tangent space to any grid node. Then we define the concepts of vector field, field of matrices and inner products on the space of grid functions and on the space of vector fields, mimickin...

Abstract. The problem of stability of difference schemes for second-order evolution problems is considered. Difference schemes are treated as abstract Cauchy problems for difference equations with operator coefficients in a Banach or Hilbert space. To construct stable difference schemes the regularization principle is employed, i.e., one starts from any simple scheme (possibly unstable) and der...