Regular Differential Operators Research Articles

On System of Root Vectors of Perturbed Regular Second-Order Differential Operator Not Possessing Basis Property

This article delves into the spectral problem associated with a multiple differentiation operator that features an integral perturbation of boundary conditions of one specific type, namely, regular but not strengthened regular. The integral perturbation is characterized by the function px, which belongs to the space L20,1. The concept of problems involving integral perturbations of boundary conditions has been the subject of previous studies, and the spectral properties of such problems have been examined in various early papers. What sets the problem under consideration apart is that the system of eigenfunctions for the unperturbed problem (when px≡0) lacks the property of forming a basis. To address this, a characteristic determinant for the spectral problem has been constructed. It has been established that the set of functions px, for which the system of eigenfunctions of the perturbed problem does not constitute an unconditional basis in L20,1, is dense within the space L20,1. Furthermore, it has been demonstrated that the adjoint operator shares a similar structure.

On the Uniform Stability of Recovering Sine-Type Functions with Asymptotically Separated Zeros

We obtain the uniform stability of recovering entire functions of special form from their zeros. To such a form, we can reduce the characteristic determinants of strongly regular differential operators and pencils of the first and the second orders, including differential systems with asymptotically separated eigenvalues whose characteristic numbers lie on a line containing the origin, as well as the nonlocal perturbations of these operators. We prove that the dependence of such functions on the sequences of their zeros is Lipschitz continuous with respect to natural metrics on each ball of a finite radius. Results of this type can be used for studying the uniform stability of inverse spectral problems. In addition, general theorems on the asymptotics of zeros of functions of this class and on their equivalent representation via an infinite product are obtained, which give the corresponding results for many specific operators.

The Friedrichs extension of regular symmetric differential operators

The Friedrichs extension of regular symmetric differential operators

On zeros of an entire function coinciding with exponential typequasi-polynomials, associated with a regular third-order differential operator on an interval

In this paper, we consider the question on study of zeros of an entire function of one class, which coincides with quasi-polynomials of exponential type. Eigenvalue problems for some classes of differential operators on a segment are reduced to a similar problem. In particular, the studied problem is led by the eigenvalue problem for a linear differential equation of the third order with regular boundary value conditions in the space W^3_2(0, 1). The studied entire function is adequately characteristic determinant of the spectral problem for a third-order linear differential operator with periodic boundary value conditions. An algorithm to construct a conjugate indicator diagram of an entire function of one class is indicated, which coincides with exponential type quasi-polynomials with comparable exponents according to the monograph by A.F. Leontyev. Existence of a countable number of zeros of the studied entire function in each series is proved, which are simultaneously eigenvalues of the above-mentioned third-order differential operator with regular boundary value conditions. We determine distance between adjacent zeros of each series, which lies on the rays perpendicular to sides of the conjugate indicator diagram, that is a regular hexagon on the complex plane. In this case, zero is not an eigenvalue of the considered operator, that is, zero is a regular point of the operator. Fundamental difference of this work is finding the corresponding eigenfunctions of the operator. System of eigenfunctions of the operator corresponding in each series is found. Adjoint operator is constructed.

AN UNBOUNDED OPERATOR WITH SPECTRUM IN A STRIP AND MATRIX DIFFERENTIAL OPERATORS

AbstractLet A and $\tilde A$ be unbounded linear operators on a Hilbert space. We consider the following problem. Let the spectrum of A lie in some horizontal strip. In which strip does the spectrum of $\tilde A$ lie, if A and $\tilde A$ are sufficiently ‘close’? We derive a sharp bound for the strip containing the spectrum of $\tilde A$ , assuming that $\tilde A-A$ is a bounded operator and A has a bounded Hermitian component. We also discuss applications of our results to regular matrix differential operators.

Fourth order differential operators with distributional potentials

In this paper, regular and singular fourth order differential operators with distributional potentials are investigated. In particular, existence and uniqueness of solutions of the fourth order differential equations are proved, deficiency indices theory of the corresponding minimal symmetric operators are studied. These symmetric operators are considered as acting on the single and direct sum Hilbert spaces. The latter one consists of three Hilbert spaces such that a squarely integrable space and two spaces of complex numbers. Moreover all maximal self-adjoint, maximal dissipative and maximal accumulative extensions of the minimal symmetric operators including direct sum operators are given in the single and direct sum Hilbert spaces.

Canonical forms of self-adjoint boundary conditions for regular differential operators of order three

Canonical forms of self-adjoint boundary conditions for regular differential operators of order three

Global regularity of second order twisted differential operators

In this paper we characterize global regularity in the sense of Shubin of twisted partial differential operators of second order in dimension 2. These operators form a class containing the twisted Laplacian, and in bi-unique correspondence with second order ordinary differential operators with polynomial coefficients and symbol of degree 2. This correspondence is established by a transformation of Wigner type. In this way the global regularity of twisted partial differential operators turns out to be equivalent to global regularity and injectivity of the corresponding ordinary differential operators, which can be completely characterized in terms of the asymptotic behavior of the Weyl symbol. In conclusion we observe that we have obtained a new class of globally regular partial differential operators which is disjoint from the class of hypo-elliptic operators in the sense of Shubin.

A Comparison Between Fractional-Order and Integer-Order Differential Finite Deformation Viscoelastic Models: Effects of Filler Content and Loading Rate on Material Parameters

Elastomers or rubber-like materials exhibit nonlinear viscoelastic behavior such as creep and relaxation upon mechanical loading. Differential constitutive models and hereditary integrals are the main frameworks followed in the literature for modeling the viscoelastic behavior at finite deformations. Regular differential operators can be replaced by fractional-order derivatives in the standard models in order to make fractional viscoelastic models. In the present paper, the relaxation behavior of elastomers is formulated both in terms of ordinary (integer-order) and fractional differential viscoelastic models. The derived constitutive equations are fitted to several experimental data to compare their efficiency in modeling the stress relaxation phenomenon. Specifically, a fractional viscoelastic model with one fractional dashpot (FD) is compared with two ordinary models including respectively one and two ordinary dashpots (OD). The models are compared in fitting accuracy, number of required material parameters and also variation of parameters from one compound to another to clarify the effects of filler content and deformation rate. It is shown that, the results of the ordinary model with one OD is not good at all. The fractional model with one FD and the ordinary model with two ODs provide good fittings for all compounds whereas the former uses only three parameters and the latter uses five material parameters. For the fractional model, the order of the Maxwell element and the associated relaxation time approximately remain the same for different compounds of each material at certain loading rates, but it is not the case for the ordinary differential models.

Application of fractional time derivatives in modeling the finite deformation viscoelastic behavior of carbon-black filled NR and SBR

Mechanical behavior of elastomers includes nonlinear and time dependent phenomena such as creep and relaxation where due to low stiffness, these materials usually undergo large deformations. The classical approaches of viscoelasticity,i.e. differential constitutive models and hereditary integrals have been widely used to model the viscoelastic behavior of these materials even at finite deformations. However, from the viewpoint of computational analysis, the integral based methods are often time and memory consuming and the differential methods usually need a large number of material parameters whose identification is a real challenge. A potential solution is to introduce fractional order derivatives instead of regular differential operators in the standard spring-dashpot mechanical models which remarkably reduces the number of required material parameters. In the present paper, relaxation behavior of NR and SBR is investigated in finite strains. Due to the remarkable effects of filler particles and deformation rate on the mechanical properties of elastomers, compounds of each material with different amounts of filler content are investigated at two strain rates. A main interest is to examine efficiency of the fractional derivative models in modeling the finite deformation relaxation behavior of elastomers with different filler contents. The other goal is to investigate the effect of loading rate on the obtained model parameters. It is observed that, the fractional Maxwell chain is able to well fit the stress relaxation response of all compounds with only three parameters. Analyzing the obtained parameters and the experimental data, it is concluded that, order of the associated Maxwell element is not much affected with the filler content. Seemingly, it depends on the matrix material and more noticeably on the deformation rate. Moreover, as much as the deformation rate, the associated relaxation time is low. Decrease of the relaxation time with the deformation rate is more evident for compounds of SBR compared with NR compounds.

A regular differential operator with perturbed boundary condition

The operator ℒ0 generated by a linear ordinary differential expression of nth order and regular boundary conditions of general form is considered on a closed interval. The characteristic determinant of the spectral problem for the operator ℒ1, where ℒ1 is an operator with the integral perturbation of one of its boundary conditions, is constructed, assuming that the unperturbed operator ℒ0 possesses a system of eigenfunctions and associated functions generating an unconditional basis in L 2(0, 1). Using the obtained formula, we derive conclusions about the stability or instability of the unconditional basis properties of the system of eigenfunctions and associated functions of the problem under an integral perturbation of the boundary condition. The Samarskii–Ionkin problem with integral perturbation of its boundary condition is used as an example of the application of the formula.

Green function of fourth-order differential operator with eigenparameter-dependent boundary and transmission conditions

We investigate a class of fourth-order regular differential operator with transmission conditions at an interior discontinuous point and the eigenparameter appears not only in the differential equation but also in the boundary conditions. We prove that the operator is symmetric, construct basic solutions of differential equation, and give the corresponding Green function of the operator is given.

Variations of Hodge Structures for Hypergeometric Differential Operators and Parabolic Higgs Bundles

Consider the holomorphic bundle with connection on $\mathbb P^1-\{0,1,\infty\}$ corresponding to the regular hypergeometric differential operator \[ \prod_{j=1}^h(D-\alpha_j)-z\prod_{j=1}^h(D-\beta_j), \qquad D=z\frac{d}{dz}. \] If the numbers $\alpha_i$ and $\beta_j$ are real and for all $i$ and $j$ the number $\alpha_i-\beta_j$ is not integer, then the bundle with connection is known to underlie a complex polarizable variation of Hodge structures. We calculate some Hodge invariants for this variation, in particular, the Hodge numbers. From this we derive a conjecture of Corti and Golyshev. We also use non-abelian Hodge theory to interpret our theorem as a statement about parabolic Higgs bundles.

The Structure of the Kac–Wang–Yan Algebra

The Lie algebra $${\mathcal{D}}$$ of regular differential operators on the circle has a universal central extension $${\hat{\mathcal{D}}}$$ . The invariant subalgebra $${\hat{\mathcal{D}}^+}$$ under an involution preserving the principal gradation was introduced by Kac, Wang, and Yan. The vacuum $${\hat{\mathcal{D}}^+}$$ -module with central charge $${c \in \mathbb{C}}$$ , and its irreducible quotient $${\mathcal{V}_c}$$ , possess vertex algebra structures, and $${\mathcal{V}_c}$$ has a nontrivial structure if and only if $${c \in \frac{1}{2}\mathbb{Z}}$$ . We show that for each integer $${n > 0}$$ , $${\mathcal{V}_{n/2}}$$ and $${\mathcal{V}_{-n}}$$ are $${\mathcal{W}}$$ -algebras of types $${\mathcal{W}(2, 4,\dots,2n)}$$ and $${\mathcal{W}(2, 4,\dots, 2n^2 + 4n)}$$ , respectively. These results are formal consequences of Weyl’s first and second fundamental theorems of invariant theory for the orthogonal group $${{\rm O}(n)}$$ and the symplectic group $${{\rm Sp}(2n)}$$ , respectively. Based on Sergeev’s theorems on the invariant theory of $${{\rm Osp}(1, 2n)}$$ we conjecture that $${\mathcal{V}_{-n+1/2}}$$ is of type $${\mathcal{W}(2, 4,\dots, 4n^2 + 8n + 2)}$$ , and we prove this for $${n = 1}$$ . As an application, we show that invariant subalgebras of $${\beta\gamma}$$ -systems and free fermion algebras under arbitrary reductive group actions are strongly finitely generated.

The super $\mathcal {W}_{1+\infty }$ algebra with integral central charge

The Lie superalgebra SD of regular differential operators on the super circle has a universal central extension \hat{SD}. For each c\in C, the vacuum module M_c(\hat{SD}) of central charge c admits a vertex superalgebra structure, and M_c(\hat{SD}) \cong M_{-c}(\hat{SD}). The irreducible quotient V_c(\hat{SD}) of the vacuum module is known as the super W_{1+\infty} algebra. We show that for each integer n>0, V_n(\hat{SD}) has a minimal strong generating set consisting of 4n fields, and we identify it with a W-algebra associated to the purely odd simple root system of gl(n|n). Finally, we realize V_n(\hat{SD}) as the limit of a family of commutant vertex algebras that generically have the same graded character and possess a minimal strong generating set of the same cardinality.

The Tamarkin equiconvergence theorem and a first-order trace formula for regular differential operators revisited

We obtain a simple formula for the first-order trace of a regular differential operator on a segment perturbated by a multiplication operator. The main analytic ingredient of the proof is an improvement of the Tamarkin equiconvergence theorem.

The exotic heat-trace asymptotics of a regular-singular operator revisited

We discuss the exotic properties of the heat-trace asymptotics for a regular-singular operator with general boundary conditions at the singular end, as observed by Falomir, Muschietti, Pisani, and Seeley [“Unusual poles of the ζ-functions for some regular singular differential operators,” J. Phys. A 36(39), 9991–10010 (2003)]10.1088/0305-4470/36/39/302 as well as by Kirsten, Loya, and Park [“The very unusual properties of the resolvent, heat kernel, and zeta function for the operator −d2/dr2 − 1./(4r2),” J. Math. Phys. 47(4), 043506 (2006)]10.1063/1.2189194. We explain how their results alternatively follow from the general heat kernel construction by Mooers [“Heat kernel asymptotics on manifolds with conic singularities,” J. Anal. Math. 78, 1–36 (1999)]10.1007/BF02791127, a natural question that has not been addressed yet, as the latter work did not elaborate explicitly on the singular structure of the heat trace expansion beyond the statement of non-polyhomogeneity of the heat kernel.

Optimal regular differential operators with variable coefficients and applications

In this paper, maximal regularity properties for linear and nonlinear high-order elliptic differential-operator equations with VMO coefficients are studied. For the linear case, the uniform coercivity property of parameter-dependent boundary value problems is obtained in spaces. Then, the existence and uniqueness of a strong solution of the boundary value problem for a high-order nonlinear equation are established. In application, the maximal regularity properties of the anisotropic elliptic equation and the system of equations with VMO coefficients are derived. AMS Subject Classification:58I10, 58I20, 35Bxx, 35Dxx, 47Hxx, 47Dxx.

Spectral properties of self-adjoint subspaces

This paper focuses on spectral properties of self-adjoint subspaces in the product space of a Hilbert space. Several classifications of the spectrum of a self-adjoint subspace are introduced, including discrete spectrum, essential spectrum, continuous spectrum, singular continuous spectrum, absolutely continuous spectrum, and singular spectrum. Their relationships between the subspace and its operator part are established, respectively. It is shown that all self-adjoint subspace extensions of a closed Hermitian subspace have the same essential spectrum. As a simple application, it is shown that every self-adjoint subspace extension of the corresponding minimal subspace to a singular second-order symmetric linear difference equation has a bounded spectrum under some conditions. This shows that there is an essential difference between spectral properties of singular symmetric difference operators and classical regular or singular symmetric differential operators.

Boundary pairs and boundary conditions for general (not necessarily definite) first‐order symmetric systems with arbitrary deficiency indices

AbstractWe investigate in the paper general (not necessarily definite) first‐order symmetric systems of differential equations in the framework of extension theory of symmetric linear relations. For this aim we first introduce the new notion of a boundary pair \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\lbrace \mathcal H_0\oplus \mathcal H_1, \Gamma \rbrace$\end{document} for A*, where A is a symmetric linear relation, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal H_0$\end{document} is a boundary Hilbert space, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal H_1$\end{document} is a subspace in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal H_0$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Gamma :A^*\rightarrow \mathcal H_0\oplus \mathcal H_1$\end{document} is a (possibly multivalued) linear mapping satisfying the abstract Green's identity. Unlike known concept of a boundary pair for A* our definition is suitable for relations A with possibly unequal deficiency indices n±(A).Next, the general symmetric system Jy′(t) − B(t)y(t) = Δ(t)f(t) on an interval \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal I=(a,b),\; -\infty \le a < b\le \infty ,$\end{document} is considered in the paper. We characterize explicitly the corresponding minimal relation \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$T_{\textit {min}}$\end{document}, which enables us to construct a special (so‐called decomposing) boundary pair \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\lbrace \mathcal H_0\oplus \mathcal H_1, \Gamma \rbrace$\end{document} for the maximal relation \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$T_{\textit {max}}$\end{document}. It turns out that the system is definite if and only if the mapping Γ is single‐valued, in which case a decomposing boundary pair turns into the decomposing boundary triplet \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Pi =\lbrace \mathcal H,\Gamma _0,\Gamma _1\rbrace$\end{document} for \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$T_{\textit {max}}$\end{document}. By using such a triplet we describe in terms of boundary conditions proper extensions of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$T_{\textit {min}}$\end{document} in the case of the regular endpoint a and arbitrary (possibly unequal) deficiency indices \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$n_\pm (T_{\textit {min}})$\end{document}. We also show that self‐adjoint decomposing boundary conditions exist only for Hamiltonian systems; moreover, we describe all such conditions in the compact form. These results are generalizations of the known results by Rofe‐Beketov on regular differential operators. Finally, we characterize all maximal dissipative and accumulative separated boundary conditions for arbitrary (not necessarily Hamiltonian) definite symmetric systems