Functions Of Differential Operators Research Articles

Inverse problem for Dirac operators with two constant delays

Abstract We study inverse spectral problems for Dirac-type functional-differential operators with two constant delays greater than two fifths the length of the interval, under Dirichlet boundary conditions. The inverse problem of recovering operators from four spectra has been studied. We consider cases when delays are greater or less than half the length of the interval. The main result of the paper refers to the proof that in both cases operators can be recovered uniquely from four spectra.

On Recovering Sturm–Liouville-Type Operators with Global Delay on Graphs from Two Spectra

We suggest a new formulation of the inverse spectral problem for second-order functional-differential operators on star-shaped graphs with global delay. The latter means that the delay, which is measured in the direction of a specific boundary vertex, called the root, propagates through the internal vertex to other edges. Now, we intend to recover the potentials from the spectra of two boundary value problems on the graph with a common set of boundary conditions at all boundary vertices except the root. For simplicity, we focus on star graphs with equal edges when the delay parameter is not less than their length. Under the assumption that the common boundary conditions are of the Robin type and they are known and pairwise linearly independent, the uniqueness theorem is proven and a constructive procedure for solving the proposed inverse problem is obtained.

Functional-Differential Operators on Geometrical Graphs with Global Delay and Inverse Spectral Problems

We suggest a new concept of functional-differential operators with constant delay on geometrical graphs that involves global delay parameter. Differential operators on graphs model various processes in many areas of science and technology. Although a vast majority of studies in this direction concern purely differential operators on graphs (often referred to as quantum graphs), recently there also appeared some considerations of nonlocal operators on star-type graphs. In particular, there belong functional-differential operators with constant delays but in a locally nonlocal version. The latter means that each edge of the graph has its own delay parameter, which does not affect any other edge. In this paper, we introduce globally nonlocal operators that are expected to be more natural for modelling nonlocal processes on graphs. We also extend this idea to arbitrary trees, which opens a wide area for further research. Another goal of the paper is to study inverse spectral problems for operators with global delay in one illustrative case by addressing a wide range of questions including uniqueness, characterization of the spectral data as well as the uniform stability.

Inverse problem for a differential operator on a star-shaped graph with nonlocal matching condition

In this paper, we develop two approaches to investigation of inverse spectral problems for a new class of nonlocal operators on metric graphs. The Laplace differential operator is considered on a star-shaped graph with nonlocal integral matching condition. This operator is adjoint to the functional-differential operator with frozen argument at the central vertex of the graph. We study the inverse problem that consists in the recovery of the integral condition coefficients from the eigenvalues. We obtain the spectrum characterization, reconstruction algorithms, and prove the uniqueness of the inverse problem solution.

Inverse Problems for Dirac Operators with Constant Delay: Uniqueness, Characterization, Uniform Stability

We initiate studying inverse spectral problems for Dirac-type functional-differential operators with constant delay. For simplicity, we restrict ourselves to the case when the delay parameter is not less than one half of the interval. For the considered case, however, we give answers to the full range of questions usually raised in the inverse spectral theory. Specifically, reconstruction of two complex $$L_{2}$$ -potentials is studied from either complete spectra or subspectra of two boundary value problems with one common boundary condition. We give conditions on the subspectra that are necessary and sufficient for the unique determination of the potentials. Moreover, necessary and sufficient conditions for the solvability of both inverse problems are obtained. For the inverse problem of recovering from the complete spectra, we establish also uniform stability in each ball of a finite radius. For this purpose, we use recent results on uniform stability of sine-type functions with asymptotically separated zeros.

Pth moment input-to-state stability of impulsive stochastic functional differential systems with Markovian switching

The paper introduces a unified criterion for pth moment input-to-state stability (p-ISS), pth moment integral input-to-state stability (p-iISS), and eσt-weighted pth moment integral input-to-state stability (eσt-p-ISS) of impulsive stochastic function differential system with Markovian switching under the perturbation of the stabilizing impulse and destabilizing impulse. The Lyapunov function approach, comparison principle, and impulsive average dwell-time method are applied in this paper. The linear coefficients of the upper bound of Lyapunov functional differential operators are time-varying functions, including the case of constants, which advances and improves the existing results. In addition, the same results were obtained by applying impulse differential inequality. At the end of this paper, we use a numerical example to verify the validity of the results.

Inverse spectral problems for functional-differential operators with involution

The main goal of this paper is to propose an approach to inverse spectral problems for functional-differential operators (FDO) with involution. For definiteness, we focus on the second-order FDO with involution-reflection. Our approach is based on the reduction of the problem to the matrix form and on the solution of the inverse problem for the matrix Sturm-Liouville operator by developing the method of spectral mappings. The obtained matrix Sturm-Liouville operator contains the weight, which causes qualitative difficulties in the study of the inverse problem. As a result, we show that the considered FDO with involution is uniquely specified by five spectra of certain regular boundary value problems.

Universal form of the Nicolai map

The nonlocal bosonic theory obtained from integrating out all anticommuting and auxiliary variables in a globally supersymmetric theory is characterized by the Nicolai map. The latter is generated by a coupling flow functional differential operator, which can be canonically constructed when the supersymmetry is realized off shell. Given any scalar superfield theory, we present a universal formula for both the Nicolai map and its inverse in terms of an ordered exponential of the integrated coupling flow operator. We demonstrate that our formula also holds for supersymmetric gauge theories.

Construction method for the Nicolai map in supersymmetric Yang–Mills theories

Recently, a universal formula for the Nicolai map in terms of a coupling flow functional differential operator was found. We present the full perturbative expansion of this operator in Yang–Mills theories where supersymmetry is realized off-shell. Given this expansion, we develop a straightforward method to compute the explicit Nicolai map to any order in the gauge coupling. Our work extends the previously known construction method from the Landau gauge to arbitrary gauges and from the gauge hypersurface to the full gauge-field configuration space. As an example, we present the map in the axial gauge to the second order.

Some Properties of Functional-Differential Operators with Involution $\nu(x)=1-x$ and Their Applications

Some Properties of Functional-Differential Operators with Involution $\nu(x)=1-x$ and Their Applications

Asymptotics of the Spectrum of a Family of Functional-Differential Operators with Integrable Potentials

We consider a higher order functional-differential operator with an integrable potential and regular separated boundary conditions. We obtain asymptotic formulas and estimates for solutions to the corresponding functional-differential equation and asymptotics of eigenvalues of the differential operator.

On an open question in recovering Sturm–Liouville-type operators with delay

In recent years, there appeared a considerable interest in the inverse spectral theory for functional-differential operators with constant delay. In particular, it is well known that specification of the spectra of two operators ℓj,j=0,1, generated by one and the same functional-differential expression −y′′(x)+q(x)y(x−a) under the boundary conditions y(0)=y(j)(π)=0 uniquely determines the complex-valued square-integrable potential q(x) vanishing on (0,a) as soon as a∈[π∕2,π). For many years, it has been a challenging open question whether this uniqueness result would remain true also when a∈(0,π∕2). Recently, a positive answer was obtained for the case a∈[2π∕5,π∕2). In this paper, we give, however, a negative answer to this question for a∈[π∕3,2π∕5) by constructing an infinite family of iso-bispectral potentials. Some discussion on a possibility of constructing a similar counterexample for other types of boundary conditions is provided, and new open questions are outlined.

A consistent variational formulation of Bishop nonlocal rods

Thick rods are employed in Nanotechnology to build modern electro mechanical systems. Design and optimization of such structures can be carried out by nonlocal continuum mechanics which is computationally convenient when compared to atomistic strategies. Bishop's kinematics is able to describe small-scale thick rods if a proper mathematical model of nonlocal elasticity is formulated to capture size effects. In all papers on the matter, nonlocal contributions are evaluated by replacing Eringen's integral convolution with the consequent (but not equivalent) differential equation governed by Helmholtz's differential operator. As notorious in integral equation theory, this replacement is possible for convolutions, defined in unbounded domains, governed by averaging kernels which are Green's functions of differential operators. Indeed, Eringen himself, in order to study nonlocal problems defined in unbounded domains, such as screw dislocations and wave propagations, suggested to replace integro-differential equations with differential conditions. A different scenario appears in Bishop rod mechanics where nonlocal integral convolutions are defined in bounded structural domains, so that Eringen's nonlocal differential equation has to be supplemented with additional boundary conditions. The objective is achieved by formulating the governing nonlocal equations by a proper variational statement. The new methodology provides an amendment of previous contributions in literature and is illustrated by investigating the elastostatic behavior of simple structural schemes. Exact solutions of Bishop rods are evaluated in terms of nonlocal parameter and cross-section gyration radius. Both hardening and softening structural responses are predictable with a suitable tuning of the parameters.

On a class of functional–differential operators satisfying the Kato conjecture

Second order elliptic differential-difference operators with degeneration in a cylinder are considered. It is proved that these operators satisfy the Kato conjecture about the square root of an operator.

Nonlocal problems and functional-differential equations: theoretical aspects and applications to mathematical modelling

This paper presents a review of results on nonlocal problems, functional-differential equations, and their applications, obtained during several last years. The following research areas are covered: the Kato square root problem for functional-differential operators, Vlasov equations and their applications to the modelling of high-temperature plasma, specific properties of differential-difference equations with incommensurable translations, degenerate functional-differential equations and their applications, functional-differential equations with contractions and extensions of independent variables, and operational methods for parabolic and elliptic functional-differential equations.

Asymptotics of the spectrum of family functional-differential operators with summable potential

Asymptotics of the spectrum of family functional-differential operators with summable potential

Dirac Operator with a Potential of Special Form and with the Periodic Boundary Conditions

We consider the Dirac operator on the interval [0, 1] with the periodic boundary conditions and with a continuous potential Q(x) whose diagonal is zero and which satisfies the condition Q(x) = QT(1−x), x ∈ [0, 1]. We establish a relationship between the spectrum of this operator and the spectra of related functional-differential operators with involution. We prove that the system of eigenfunctions of this Dirac operator has the Riesz basis property in the space L 2 2 [0, 1].

An inverse problem for Sturm–Liouville differential operators with deviating argument

We consider second-order functional differential operators with a constant delay. Properties of their spectral characteristics are obtained and a nonlinear inverse problem is studied, which consists in recovering the operators from their spectra. We establish the uniqueness and develop a constructive algorithm for solution of the inverse problem.

On discreteness of spectrum of a second order functional differential operator

On discreteness of spectrum of a second order functional differential operator

Boundary-value problems for elliptic functional-differential equations and their applications

Boundary-value problems are considered for strongly elliptic functional-differential equations in bounded domains. In contrast to the case of elliptic differential equations, smoothness of generalized solutions of such problems can be violated in the interior of the domain and may be preserved only on some subdomains, and the symbol of a self-adjoint semibounded functional-differential operator can change sign. Both necessary and sufficient conditions are obtained for the validity of a Gårding-type inequality in algebraic form. Spectral properties of strongly elliptic functional-differential operators are studied, and theorems are proved on smoothness of generalized solutions in certain subdomains and on preservation of smoothness on the boundaries of neighbouring subdomains. Applications of these results are found to the theory of non-local elliptic problems, to the Kato square-root problem for an operator, to elasticity theory, and to problems in non-linear optics. Bibliography: 137 titles.