Differential-difference Operators Research Articles

Uniqueness of meromorphic functions concerning their differential-difference operators

Uniqueness of meromorphic functions concerning their differential-difference operators

On Global Solutions of Hyperbolic Equations with Positive Coefficients at Nonlocal Potentials

We study hyperbolic equations with positive coefficients at potentials undergoing translations with respect to the spatial independent variable. The qualitative novelty of the investigation is that the real part of the symbol of the differential-difference operator contained in the equation is allowed to change its sign. Earlier, only the case where the said sign is constant was investigated. We find a condition relating the coefficient at the nonlocal term of the investigated equation and the length of the translation, guaranteeing the global solvability of the investigated equation. Under this condition, we explicitly construct a three-parametric family of smooth global solutions of the investigated equation.

On the Construction of a Variational Principle for a Certain Class of Differential-Difference Operator Equations

On the Construction of a Variational Principle for a Certain Class of Differential-Difference Operator Equations

On the existence of time-periodic solutions of nonlinear parabolic differential equations with nonlocal boundary conditions of the Bitsadze-Samarskii type

We study a nonlinear parabolic differential equation in a bounded multidimensional domain with nonlocal boundary conditions of the Bitsadze-Samarskii type. We prove existence theorems for a periodic in time generalized solution. Su cient conditions for the existence of generalized solutions contain either an algebraic ellipticity condition or an algebraic strong ellipticity condition for the auxiliary differential-difference operator.

The Fractional Dunkl Laplacian: Definition and Harmonization via the Mellin Transform

In this paper, we extend the scope of the Tate and Ormerod Lemmas to the Dunkl setting, revealing a profound interconnection that intricately links the Dunkl transform and the Mellin transform. This illumination underscores the pivotal significance of the Mellin integral transform in the realm of fractional calculus associated with differential-difference operators. Our primary focus centers on the Dunkl–Laplace operator, which serves as a prototype of a differential-difference second-order operator within an unbounded domain. Following influential research by Pagnini and Runfola, we embark on an innovative exploration employing Bochner subordination approaches tailored for the fractional Dunkl Laplacian (FDL). Notably, the Mellin transform emerges as a robust and enlightening tool, particularly in its application to the FDL.

On the Big Hartley transform

In this paper, we introduce a new type of singular first-order differential-difference operator of Dunkl type on the real line. This operator is obtained as a limiting case from both the first-order Dunkl-type operators corresponding to Bannai-Ito and Big 1-Jacobi orthogonal polynomials. We provide an explicit expression for the eigenfunction of this operator in terms of Bessel functions. The obtained kernel is called the Big Hartley function, which is a generalization of the usual Hartley kernel and the little Hartley function studied in Bouzeffour [The generalized Hartley transform. Integral Transforms Spec Funct. 2014;25(3):230–239]. Additionally, we present a new product formula for the little Hartley function, which is related to the Kingman-Bessel hypergroup and the Rosler-Dunkl signed hypergroup. Finally, we investigate inversion formulae for the transforms of both the little Hartley function and the big Hartley function.

Resolvent Convergence for Differential–Difference Operators with Small Variable Translations

We consider general higher-order matrix elliptic differential–difference operators in arbitrary domains with small variable translations in lower-order terms. The operators are introduced by means of general higher-order quadratic forms on arbitrary domains. Each lower-order term depends on its own translation and all translations are governed by a small multi-dimensional parameter. The operators are considered either on the entire space or an arbitrary multi-dimensional domain with a regular boundary. The boundary conditions are also arbitrary and general and involve small variable translations. Our main results state that the considered operators converge in the norm resolvent sense to ones with zero translations in the best possible operator norm. Estimates for the convergence rates are established as well. We also prove the convergence of the spectra and pseudospectra.

On Solvability of Parabolic Equations with Essentially Nonlinear Differential-Difference Operators

On Solvability of Parabolic Equations with Essentially Nonlinear Differential-Difference Operators

Classical Solutions of Hyperbolic Equation with Translation Operators in Free Terms

In this paper, we study the question of constructing explicit solutions in a half-space of a hyperbolic equation containing translation operators in space variables in all coordinate directions. Such equations are a natural generalization of classical equations of hyperbolic type, and the resulting solution relates the value of the desired function at different points of the half-space where the process takes place. To construct solutions, a classical operating scheme is used, namely, the formal application of an integral transformation. A theorem is proved that the constructed solutions are classical if the real part of the symbol of the differential-difference operator in the equation is positive. Classes of equations for which this condition is satisfied are given.

Dirichlet and Neumann Boundary Value Problems for Dunkl Polyharmonic Equations

Dunkl operators are a family of commuting differential–difference operators associated with a finite reflection group. These operators play a key role in the area of harmonic analysis and theory of spherical functions. We study the solution of the inhomogeneous Dunkl polyharmonic equation based on the solutions of Dunkl–Possion equations. Furthermore, we construct the solutions of Dirichlet and Neumann boundary value problems for Dunkl polyharmonic equations without invoking the Green’s function.

The B<sub>2</sub> Harmonic Oscillator with Reflections and Superintegrability

The two-dimensional quantum harmonic oscillator is modified with reflection terms associated with the action of the Coxeter group B<sub>2</sub>, which is the symmetry group of the square. The angular momentum operator is also modified with reflections. The wavefunctions are known to be built up from Jacobi and Laguerre polynomials. This paper introduces a fourth-order differential-difference operator commuting with the Hamiltonian but not with the angular momentum operator; a specific instance of superintegrability. The action of the operator on the usual orthogonal basis of wavefunctions is explicitly described. The wavefunctions are classified according to the representations of the group: four of degree one and one of degree two. The identity representation encompasses the wavefunctions invariant under the group. The paper begins with a short discussion of the modified Hamiltonians associated to finite reflection groups, and related raising and lowering operators. In particular, the Hamiltonian for the symmetric groups describes the Calogero-Sutherland model of identical particles on the line with harmonic confinement.

Spectral Properties of Differential–Difference Symmetrized Operators

We investigate some spectral properties of differential–difference operators, which are symmetrizations of differential operators of the form (mathfrak {d}^dagger mathfrak {d})^k and (mathfrak {d}mathfrak {d}^dagger )^k, kgeqslant 1. Here, mathfrak {d}=pfrac{d}{textrm{d}x} +q and mathfrak {d}^dagger stands for the formal adjoint of mathfrak {d} on L^2((0,b),w,textrm{d}x). In the simpliest case k=1, this symmetrization brings in the operator -mathfrak {D}^2, which can be seen as a ‘Laplacian’, and mathfrak {D}f:=mathfrak {D}_{mathfrak {d}}f= mathfrak {d}(f_{text {even}})-mathfrak {d}^dagger (f_{text {odd}}), a skew-symmetric operator in L^2(I,w,textrm{d}x), I=(-,b,0)cup (0,b), is the symmetrization of mathfrak {d}. Investigated spectral properties include self-adjoint extensions, among them the Friedrichs extensions, of the symmetrized operators.

Harmonic Analysis Associated with the Jacobi–Dunkl Operator on (-pi ,pi ): Exotic Cases

We investigate some spectral properties of a second order differential-difference operator J_{alpha ,beta } on L^2((-pi ,pi ),dmu _{alpha , beta }), alpha ,beta in mathbb {R}, called the Jacobi–Dunkl operator of compact type. Using an idea of Hajmirzaahmad, in exotic cases, e.g. when at least one of the two parameters alpha ,beta is le -1, we construct exotic orthonormal bases that consist of eigenfunctions of J_{alpha ,beta }. This allows one to consider natural self-adjoint exotic extensions of J_{alpha ,beta } and the corresponding exotic Jacobi–Dunkl and Jacobi–Dunkl–Poisson semigroups.

Initial Problem for Two-Dimensional Hyperbolic Equation with a Nonlocal Term

In this paper, we study the Cauchy problem in a strip for a two-dimensional hyperbolic equation containing the sum of a differential operator and a shift operator acting on a spatial variable that varies over the real axis. An operating scheme is used to construct the solutions of the equation. The solution of the problem is obtained in the form of a convolution of the function found using the operating scheme and the function from the initial conditions of the problem. It is proved that classical solutions of the considered initial problem exist if the real part of the symbol of the differential-difference operator in the equation is positive.

Index of Differential-Difference Operators on an Infinite Cylinder

Index of Differential-Difference Operators on an Infinite Cylinder

Correction to: Unicity on Entire Function Concerning Its Differential-Difference Operators

In this paper, we give a correction to Theorem 1 in a previous paper [Results Math. 76, Article number: 147 (2021)].

Flett potentials associated with differential-difference Laplace operators

A large family of Flett potentials is investigated. Formally, these potentials are negative powers of the operators id + |x|1−1/mΔk, where Δk is the Dunkl Laplace (differential and difference) operator on R. Here, k ≥ 0 and m∈N\{0}. In the (k = 0, m = 1) case, our family of potentials reduces to the classical one studied by Flett [Proc. London Math. Soc. s3-22, 385–451 (1971)]. An explicit inversion formula of the Flett potentials is obtained for functions belonging to C0(R) and weighted Lp spaces, 1 ≤ p < ∞. As a tool, we use a wavelet-like transforms generated by a Poisson type semigroup and signed Borel measures. In this context, a fundamental theorem proving an almost everywhere convergence of a convolution operator for an approximate identity was given. The k = 0 case is already new.

Estimates of Solutions of Elliptic Differential-Difference Equations with Degeneration

We consider a second-order differential-difference equation in a bounded domain Q ⊂ ℝn. We assume that the differential-difference operator contains difference operators with degenerations corresponding to the differential operators. Also, it is assumed that the considered differential-difference operator cannot be expressed by a composition of a difference operator and a strongly elliptic differential operator. The presence of degenerate difference operators does not allow us to obtain the Gårding inequality. We prove a priori estimates implying that the considered differential-difference operator is sectorial and its Friedrichs extension exists. These estimates can be applied to study the spectrum of the Friedrichs extension as well. It is well known that elliptic differential-difference equations may have solutions that do not belong even to the Sobolev space $$ {W}_2^1(Q) $$ . However, using the obtained estimates, we prove smoothness of solutions at least in subdomains Qr generated by translations of the boundary, where $$ \underset{r}{\cup}\overline{Q_r}=\overline{Q} $$ .

On the Construction of a Variational Principle for a Certain Class of Differential-Difference Operator Equations

In this paper, we obtain necessary and sufficient conditions for the existence of variational principles for a given first-order differential-difference operator equation with a special form of the linear operator P(t) depending on t and the nonlinear operator Q. Under the corresponding conditions the functional is constructed. These conditions are obtained thanks to the well-known criterion of potentiality. Examples show how the inverse problem of the calculus of variations is constructed for given differentialdifference operators.

The heat equation on the finite Poincaré upper half-plane

A differential-difference operator is used to model the heat equation on a finite graph analogue of Poincaré’s upper half-plane. Finite analogues of the classical theta functions are shown to be solutions to the heat equation in this setting.