Singular Differential Operators Research Articles

On generalized Hardy spaces associated with singular partial differential operators

We define and study the Hardy spaces associated with singular partial differential operators. Also, a characterization by mean of atomic decomposition is investigated.

Beckner Logarithmic Uncertainty Principle for the Stockwell Transform Associated with the Singular Partial Differential Operators

Beckner Logarithmic Uncertainty Principle for the Stockwell Transform Associated with the Singular Partial Differential Operators

Spectral Properties of a Singular Differential Operator on an Interval with Transmission Conditions

Spectral Properties of a Singular Differential Operator on an Interval with Transmission Conditions

Friedrichs extension of singular symmetric differential operators

For singular even order symmetric differential operators we find the matrices which determine all symmetric extensions of the minimal operator. And for each of these symmetric operators which is bounded below we find the boundary condition of its Friedrichs extension. The operators of regular problems are bounded below and thus each one has a symmetric extension and thus its symmetric extension has a Friedrichs extension. See also https://ejde.math.txstate.edu/special/02/b1/abstr.html

Duality of generalized Hardy and BMO spaces associated with singular partial differential operator

Duality of generalized Hardy and BMO spaces associated with singular partial differential operator

Regular approximation of singular third-order differential operators

In this paper, regular approximation of third-order differential operators with two singular endpoints is investigated. Firstly the induced restriction operator is constructed and proved to be a self-adjoint operator in the Hilbert space. Moreover the convergence of operators and spectrum is obtained by strong resolvent convergence. In particular, norm resolvent convergence is given when the two endpoints are both limit circle.

SMOOTHNESS OF SOLUTIONS (SEPARABILITY) OF THE NONLINEAR STATIONARY SCHR¨ODINGER EQUATION

The equation of motion of a microparticle in various force fields is the Schr¨odinger wave equation.Many questions of quantum mechanics, in particular the thermal radiation of electromagneticwaves, lead to the problem of separability of singular differential operators. One such operator isthe above Schr?dinger operator. In this paper, the named operator is studied by the methods offunctional analysis. Found sufficient conditions for the existence of a solution and the separabilityof an operator in a Hilbert space. All theorems were originally proved for the model Sturm-Liouvilleequation and extended to a more general case.In §1-2, for the nonlinear Sturm-Liouville equation, sufficient conditions are found that ensurethe existence of an estimate for coercivity, and estimates of weight norms are obtained for thefirst derivative of the solution. In Sections 3-4 the results of Sections 1-2 are generalized for theSchr¨odinger equation in the case m = 3.

ОПРЕДЕЛЕНИЕ НЕПРЕРЫВНОГО ЗАПАЗДЫВАНИЯ В СПЕКТРАЛЬНОЙ ЗАДАЧЕ ДЛЯ ОПЕРАТОРА ЧЕБЫШЁВА ПЕРВОГО РОДА

A perturbed singular ordinary differential Chebyshev operator of the first kind with continuous delay is considered in this paper. For an arbitrary numerical sequence that does not differ much from eigenvalues sequence of an unperturbed operator, a problem is set to find the perturbation operator containing a continuous delay. A theorem of the existence of such an operator is being proved. An algorithm of finding the delay function in the form of a Fourier series is built and substantiated. The algorithm substantiation is based on the regularized traces theory.

Distribution dependent SDEs for Navier-Stokes type equations

To characterize Navier-Stokes type equations where the Laplacian is extended to a singular second order differential operator, we propose a class of SDEs depending on the distribution in future. The well-posedness and regularity estimates are derived for these SDEs.

Volterra-Type Integro-Differential Equations with Two-Point Singular Differential Operator

Volterra-Type Integro-Differential Equations with Two-Point Singular Differential Operator

Fractional linear maps in general relativity and quantum mechanics

This paper studies the nature of fractional linear transformations in a general relativity context as well as in a quantum theoretical framework. Two features are found to deserve special attention: the first is the possibility of separating the limit-point condition at infinity into loxodromic, hyperbolic, parabolic and elliptic cases. This is useful in a context in which one wants to look for a correspondence between essentially self-adjoint spherically symmetric Hamiltonians of quantum physics and the theory of Bondi–Metzner–Sachs transformations in general relativity. The analogy therefore arising suggests that further investigations might be performed for a theory in which the role of fractional linear maps is viewed as a bridge between the quantum theory and general relativity. The second aspect to point out is the possibility of interpreting the limit-point condition at both ends of the positive real line, for a second-order singular differential operator, which occurs frequently in applied quantum mechanics, as the limiting procedure arising from a very particular Kleinian group which is the hyperbolic cyclic group. In this framework, this work finds that a consistent system of equations can be derived and studied. Hence, one is led to consider the entire transcendental functions, from which it is possible to construct a fundamental system of solutions of a second-order differential equation with singular behavior at both ends of the positive real line, which in turn satisfy the limit-point conditions. Further developments in this direction might also be obtained by constructing a fundamental system of solutions and then deriving the differential equation whose solutions are the independent system first obtained. This guarantees two important properties at the same time: the essential self-adjointness of a second-order differential operator and the existence of a conserved quantity which is an automorphic function for the cyclic group chosen.

Essential spectra of singular Hamiltonian differential operators of arbitrary order under a class of perturbations

AbstractThe main object of this paper is to study the essential spectrum of a Hamiltonian system of arbitrary order with one singular endpoint under a class of perturbations. We first present a characterization of the essential spectrum in terms of singular sequences and then give the concept of perturbations small at singular endpoints of Hamiltonian systems. Based on the above characterization, the invariance of essential spectra of Hamiltonian systems under these perturbations is shown. It is noted that these perturbations are given by using the associated pre‐minimal operator , which provides great convenience in the study of essential spectra of Hamiltonian systems since each element of the domain of has compact support. As applications, some sufficient conditions for the invariance of essential spectra of some systems are obtained in terms of coefficients of systems and perturbations terms. Further, essential spectra of Hamiltonian systems with different weight functions are discussed. Here, Hamiltonian systems may be non‐symmetric.

Analysis of blood liquor model via nonlocal and singular constant proportional Caputo hybrid differential operator

In this study, a physical scheme called the blood liquor absorption model has been examined in its fractional (non‐integer) edict form. The constant proportional Caputo (CPC) hybrid fractional operator with singular and nonlocal kernel has been used to fractionalize the blood alcohol model. The logical solutions of the absorptions of liquor in stomach S(t) and the absorptions of liquor in the blood B(t) have been explored by using the Laplace transform technique and are stated in the forms of the generalized G‐functions Ga, b, c(·) and the bivariate Mittage–Leffler functions. Further, we present a detailed analysis including numerical explanation and stability analysis.

Kipriyanov–Beltrami Operator with Negative Dimension of the Bessel Operators and the Singular Dirichlet Problem for the $$B $$-Harmonic Equation

For the Kipriyanov operator $$\Delta _B$$ written in the form of the sum of singular differential Bessel operators with, generally speaking, negative parameters, we obtain a representation in spherical coordinates (the Kipriyanov–Beltrami operator). The operator $$\Delta _B$$ on the sphere and the corresponding spherical functions ( $$B $$ -harmonics) are introduced. The main properties of the operator $$ \Delta _B$$ on the sphere and the differential equation of $$B $$ -harmonics are provided. A solution of the inner singular Dirichlet problem in a ball centered at the origin in $$\mathbb {R}^n $$ is given. The solution is obtained by the Fourier method in the form of Laplace series in $$B$$ -harmonics. The solution is bounded only in the case where all parameters of the Bessel operators occurring in $$\Delta _B $$ belong to the interval $$(-1,2/n-1) $$ , $$n\in \mathbb {N}$$ , $$n\geq 3 $$ .

Direct approach for the characteristic function of a dissipative operator with distributional potentials

The main aim of this paper is to investigate the spectral properties of a singular dissipative differential operator with the help of its Cayley transform. It is shown that the Cayley transform of the dissipative differential operator is a completely non-unitary contraction with finite defect indices belonging to the class $$C_{0}.$$ Using its characteristic function and the spectral properties of the resolvent operator, the complete spectral analysis of the dissipative differential operator is obtained. Embedding the Cayley transform to its natural unitary colligation, a Caratheodory function is obtained. Moreover, the truncated CMV matrix is established which is unitary equivalent to the Cayley transform of the dissipative differential operator. Furthermore, it is proved that the imaginary part of the inverse operator of the dissipative differential operator is a rank-one operator and the model operator of the associated dissipative integral operator is constructed as a semi-infinite triangular matrix. Using the characteristic function of the dissipative integral operator with rank-one imaginary component, associated Weyl functions are established.

First order maximally dissipative singular differential operators

In this paper, using the Calkin-Gorbachuk method, the general form of all maximal dissipative extensions of the minimal operator generated by first order linear multipoint symmetric singular differential-operator expression in the direct sum of Hilbert space of vector-functions has been found. Later on, the structure of spectrum of these extensions is researched. Finally, the results are supported by an application.

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.

Complete transformation of Radon-Kipriyanov. Some properties

The even Radon-Kipriyanov transform (Kg-transform) is suitable for investigating problems with the Bessel singular differential operator Bi = 2i2+iii,i 0. In this paper, we introduce the odd Radon-Kipriyanov transform and complete Radon-Kipriyanov transform to investigation more general equations containing odd B‑derivativesiBik, k = 0, 1, 2, ... (in particular, gradients of functions). Formulas of K-transforms of singular differential operators are given. Based on the Bessel transforms introduced by B. M. Levitan and the odd Bessel transform introduced by I. A. Kipriyanov and V. V. Katrakhov, a connection was obtained between the complete Radon-Kipriyanov transform with the Fourier transform and the mixed Fourier-Levitan-Kipriyanov-Katrakhov transform. An analogue of Helgasons support theorem and an analogue of the Paley-Wiener theorem are presented.

Complete Radon–Kipriyanov Transform: Some Properties

The even Radon–Kipriyanov transform (Kγ-transform) is suitable for studying problems with the Bessel singular differential operator $${{B}_{{{{\gamma }_{i}}}}} = \frac{{{{\partial }^{2}}}}{{\partial x_{i}^{2}}} + \frac{{{{\gamma }_{i}}}}{{{{x}_{i}}~}}\frac{\partial }{{\partial {{x}_{i}}}},{{\gamma }_{i}} > 0$$. In this work, the odd Radon–Kipriyanov transform and the complete Radon–Kipriyanov transform are introduced to study more general equations containing odd B-derivatives $$\frac{\partial }{{\partial {{x}_{i}}}}~B_{{{{\gamma }_{i}}}}^{k},~~k = 0, 1, 2,~ \ldots $$ (in particular, gradients of functions). Formulas of the Kγ-transforms of singular differential operators are given. Based on the Bessel transforms introduced by B.M. Levitan and the odd Bessel transform introduced by I.A. Kipriyanov and V.V. Katrakhov, a relationship of the complete Radon–Kipriyanov transform with the Fourier transform and the mixed Fourier–Levitan–Kipriyanov–Katrakhov transform is deduced. An analogue of Helgason’s support theorem and an analog of the Paley–Wiener theorem are given.

Convergence of Spectral Decompositions for a Singular Differential Operator with General Boundary Conditions

We investigate the general boundary-value problem for the operator lu = −u′′ + q(x)u , 0 < x < 1, If the complex-valued coefficients q(x) is summable on (0,1), the integral $$ {\int}_0^1x\left(1-x\right)\left|q(x)\right| dx $$ converges. The asymptotic solutions of the equation lu = μ2u derived in this article are used to construct the asymptotic spectrum of the problem, to classify the boundary conditions, and to prove theorems asserting that the system of root functions is complete and forms an unconditional basis in L2 (0,1).