Related papers: Discrete analogues of the Dixmier operators

Different finite difference replacements for the derivative are analyzed in the context of the Heisenberg commutation relation. The type of the finite difference operator is shown to be tied to whether one can naturally consider $P$ and $X$…

We introduce the notion of discrete Baker-Akhiezer (DBA) modules, which are modules over the ring of difference operators, as a certain discretization of Baker-Akhiezer modules which are modules over the ring of differential operators. We…

We consider one-point commuting difference operators of rank one. The coefficients of these operators depend on a functional parameter, shift operators being included only with positive degrees. We study these operators in the case of…

There is a commutative algebra of differential-difference operators, with two parameters, associated to any dihedral group with an even number of reflections. The intertwining operator relates this algebra to the algebra of partial…

A classification of commutative integral domains consisting of ordinary differential operators with matrix coefficients is established in terms of morphisms between algebraic curves.

In the present paper the discrete analogue of the differential operator $\frac{d^4}{dx^4}+2\frac{d^2}{dx^2}+1$ is constructed and its some properties are proved.

For a class of zero order pseudodifferential operators we find the asymptotics of eigenvalues converging to a non-isolated tip of the essential spectrum.

We construct new examples of multidimensional commuting matrix differential operators and a multidimensional analog of the Kadomtsev--Petviashvili hierarchy.

We consider the discrete analogue of a fractional integral operator on the Heisenberg group, for which we are able to prove nearly sharp results by means of a simple argument of a combinatorial nature.

We present explicit generators of an algebra of commuting difference operators with trigonometric coefficients. The operators are simultaneously diagonalized by recently discovered q-polynomials (viz. Koornwinder's multivariable…

We give new examples of linear differential operators of order $k=2m+1$ (any given odd integer) that are invariant under the isometries of $\mathbb R^n$ and satisfy so-called $L^1$-duality estimates and div/curl inequalities.

In this paper we study commuting difference operators containing a shift operator with only positive degrees. We construct examples of such operators in the case of hyperelliptic spectral curves.

We associate to an integral operator a discrete one which is conceptually simpler, and study the relations between them.

Representations of polynomial covariance commutation relations by pairs of linear integral and differential operators are constructed in the space of infinitely continuously differentiable functions. Representations of polynomial covariance…

Divided difference operators are degree-reducing operators on the cohomology of flag varieties that are used to compute algebraic invariants of the ring (for instance, structure constants). We identify divided difference operators on the…

We consider the self-adjoint fourth-order operator with real $1$-periodic coefficients on the unit interval. The spectrum of this operator is discrete. We determine the high energy asymptotics for its eigenvalues.

In this paper we study self-adjoint commuting ordinary differential operators. We find sufficient conditions when an operator of fourth order commuting with an operator of order $4g+2$ is self-adjoint. We introduce an equation on…

We introduce and systematically develop two classes of discrete integrable operators: those with $2\times 2$ matrix kernels and those possessing general differential kernels, thereby generalizing the discrete analogue previously studied. A…

We construct a commutative algebra A_z, generated by d algebraically independent q-difference operators acting on variables z_1, z_2,..., z_d, which is diagonalized by the multivariable Askey-Wilson polynomials P_n(z) considered by Gasper…

We propose the assumption of quantum mechanics on a discrete space and time, which implies the modification of mathematical expressions for some postulates of quantum mechanics. In particular we have a Hilbert space where the vectors are…