Liouville Operator Research Articles

On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives

In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann–Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work.

Random Sturm–Liouville operators with point interactions

AbstractWe study invariance for eigenvalues of selfadjoint Sturm–Liouville operators with local point interactions. Such linear transformations are formally defined by or similar expressions with instead of δ. In a probabilistic setting, we show that a point is either an eigenvalue for all ω or only for a set of ω's of measure zero. Using classical oscillation theory it is possible to decide whether the second situation happens. The operators do not need to be measurable or ergodic. This generalizes the well known fact that for ergodic operators a point is eigenvalue with probability zero.

Tensor method for optimal control problems constrained by fractional three‐dimensional elliptic operator with variable coefficients

AbstractWe introduce the tensor numerical method for solving optimal control problems that are constrained by fractional two‐ (2D) and three‐dimensional (3D) elliptic operators with variable coefficients. We solve the governing equation for the control function which includes a sum of the fractional operator and its inverse, both discretized over large 3D spacial grids. Using the diagonalization of the arising matrix‐valued functions in the eigenbasis of the one‐dimensional Sturm–Liouville operators, we construct the rank‐structured tensor approximation with controllable precision for the discretized fractional elliptic operators and the respective preconditioner. The right‐hand side in the constraining equation (the optimal design function) is supposed to be represented in a form of a low‐rank canonical tensor. Then the equation for the control function is solved in a tensor structured format by using preconditioned CG iteration with the adaptive rank truncation procedure that also ensures the accuracy of calculations, given an ‐threshold. This method reduces the numerical cost for solving the control problem to (plus the quadratic term with a small weight), which outperforms traditional approaches with complexity in the 3D case. The storage for the representation of all 3D nonlocal operators and functions involved is also estimated by . This essentially outperforms the traditional methods operating with fully populated matrices and vectors in . Numerical tests for 2D/3D control problems indicate the almost linear complexity scaling of the rank truncated preconditioned conjugate gradient iteration in the univariate grid size n.

Inverse problem solution and spectral data characterization for the matrix Sturm–Liouville operator with singular potential

The self-adjoint matrix Sturm–Liouville operator on a finite interval with singular potential of class $$W_2^{-1}$$ and the general self-adjoint boundary conditions is studied. This operator generalizes the Sturm–Liouville operators on geometrical graphs. We investigate the inverse problem that consists in recovering the considered operator from the spectral data (eigenvalues and weight matrices). The inverse problem is reduced to a linear equation in a suitable Banach space, and a constructive algorithm for the inverse problem solution is developed. Moreover, we obtain the spectral data characterization for the studied operator. In addition, the main results are applied to the Sturm–Liouville operator on a graph of arbitrary geometrical structure.

On Solvability of the Sonin–Abel Equation in the Weighted Lebesgue Space

In this paper we present a method of studying a convolution operator under the Sonin conditions imposed on the kernel. The particular case of the Sonin kernel is a kernel of the fractional integral Riemman–Liouville operator, other various types of the Sonin kernels are a Bessel-type function, functions with power-logarithmic singularities at the origin e.t.c. We pay special attention to study kernels close to power type functions. The main our aim is to study the Sonin–Abel equation in the weighted Lebesgue space, the used method allows us to formulate a criterion of existence and uniqueness of the solution and classify a solution, due to the asymptotics of the Jacobi series coefficients of the right-hand side.

An inverse spectral problem for second-order functional-differential pencils with two delays

Recently, there appeared a considerable interest in inverse Sturm–Liouville-type problems with constant delay. However, necessary and sufficient conditions for solvability of such problems were obtained only in one very particular situation. Here we address this gap by obtaining necessary and sufficient conditions in the case of functional-differential pencils possessing a more general form along with a nonlinear dependence on the spectral parameter. For this purpose, we develop the so-called transformation operator approach, which allows reducing the inverse problem to a nonlinear vectorial integral equation. In Appendix A, we obtain as a corollary the analogous result for Sturm–Liouville operators with delay. Remarkably, the present paper is the first work dealing with an inverse problem for functional-differential pencils in any form. Besides generality of the pencils under consideration, an important advantage of studying the inverse problem for them is the possibility of recovering both delayed terms, which is impossible for the Sturm–Liouville operators with two delays. The latter, in turn, is illustrated even for different values of these two delays by a counterexample in Appendix B. We also provide a brief survey on the contemporary state of the inverse spectral theory for operators with delay observing recently answered long-term open questions.

Identification of the Domain of the Sturm–Liouville Operator on a Star Graph

This article is devoted to the unique recovering of the domain of the Sturm–Liouville operator on a star graph. The domain of the Sturm–Liouville operator is uniquely identified from the set of spectra of a finite number of specially selected canonical problems. In the general case, the domain of the definition of the original operator can be specified by integro-differential linear forms. In the case when the domain of the Sturm–Liouville operator on a star graph corresponds to the boundary value problem, it is sufficient to choose only finite parts of the spectra of canonical problems for a unique identification of the boundary form. Moreover, the above statement is valid only for a symmetric star graph.

The Krein–von Neumann extension revisited

We revisit the Krein–von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein–von Neumann extension for singular, general (i.e., three-coefficient) Sturm–Liouville operators on arbitrary intervals. In particular, the boundary conditions for the Krein–von Neumann extension of the strictly positive minimal Sturm–Liouville operator are explicitly expressed in terms of generalized boundary values adapted to the (possible) singularity structure of the coefficients near an interval endpoint.

Application of the Fractional Sturm–Liouville Theory to a Fractional Sturm–Liouville Telegraph Equation

In this paper, we consider a non-homogeneous time–space-fractional telegraph equation in n-dimensions, which is obtained from the standard telegraph equation by replacing the first- and second-order time derivatives by Caputo fractional derivatives of corresponding fractional orders, and the Laplacian operator by a fractional Sturm–Liouville operator defined in terms of right and left fractional Riemann–Liouville derivatives. Using the method of separation of variables, we derive series representations of the solution in terms of Wright functions, for the homogeneous and non-homogeneous cases. The convergence of the series solutions is studied by using well known properties of the Wright function. We show also that our series can be written using the bivariate Mittag-Leffler function. In the end of the paper some illustrative examples are presented.

Existence criteria and solution search by the analytic technique of functional integral equation

Existence of a solution of the functional integral equation in an unbounded interval involving the Riemann–Liouville operator is investigated. Here sufficient conditions in the context of existence and stability are derived by employing hybridized fixed point theory in the Banach algebra setting. Further, an example is presented to showcase the validity of the obtained result. Moreover, the solution of the example in closed form is estimated by the semianalytic technique which is being driven by a modified homotopy perturbation method in conjunction with the Adomian decomposition method.

On non-uniqueness of recovering Sturm–Liouville operators with delay

The inverse spectral theory was developed most widely for purely differential operators. Meanwhile, various their nonlocal perturbations are often more adequate for modelling various real-world processes frequently possessing a nonlocal nature. Thus, there recently appeared a growing interest in the nonlinear inverse problems for differential operators with constant delay. However, because of their peculiarity, such problems are still insufficiently studied. In particular, there has been a long-term basic open question about the unique solvability of such inverse problems for small values of the delay. In this paper, we address that question by giving a negative answer in an important and illustrative situation. Our study is based on numerical simulations, which have finally led to a deep theoretical result. Moreover, we develop a general approach for constructing relevant iso-bispectral potentials. The obtained results essentially change the future strategy of studying inverse problems for differential operators with delay.

Extensions of the matrix-valued q−Sturm–Liouville operators

In this paper, we investigate the matrix-valued$\ q-$Sturm--Liouville problems. We establish an existence and uniqueness result. Later, we introduce the corresponding maximal and minimal operators for this system. Moreover, we give a criterion under which these operators are self-adjoint. Finally, we characterize extensions (maximal dissipative, maximal accumulative, and self-adjoint) of the minimal symmetric operator.

Eikonal solutions for moment hierarchies of chemical reaction networks in the limits of large particle number

Trajectory-based methods are well-developed to approximate steady-state probability distributions for stochastic processes in large-system limits. The trajectories are solutions to equations of motion of Hamiltonian dynamical systems, and are known as eikonals. They also express the leading flow lines along which probability currents balance. The existing eikonal methods for discrete-state processes including chemical reaction networks are based on the Liouville operator that evolves generating functions of the underlying probability distribution. We have previously derived [, ] a representation for the generators of such processes that acts directly on the hierarchy of moments of the distribution, rather than on the distribution itself or on its generating function. We show here how in the large-system limit the steady-state condition for that generator reduces to a mapping from eikonals to the ratios of neighboring factorial moments, as a function of the order k of these moments. The construction shows that the boundary values for the moment hierarchy, and thus its whole solution, are anchored in the interior fixed points of the Hamiltonian system, a result familiar from Freidlin–Wenztell theory. The direct derivation of eikonals from the moment representation further illustrates the relation between coherent-state and number fields in Doi–Peliti theory, clarifying the role of canonical transformations in that theory.

The Schatten–von Neumann class associated with the Gabor–Riemann–Liouville operator

In this paper, we define the localization operator associated with the Riemann–Liouville operator, and show that it is not only bounded, but it is also in the Schatten–von Neumann class. We also give a trace formula when the symbol function is nonnegative.

Solving fractional Fredholm integro–differential equations using Legendre wavelets

This paper presents a Legendre wavelet spectral method for solving a type of fractional Fredholm integro–differential equations. The fractional derivative is defined in the Caputo–Prabhakar sense. The derivative of Prabhakar consists of an integro–differential operator that has a Mittag–Leffler function with three parameters in the integration kernel, so it generalizes the Riemann–Liouville and Caputo fractional operators. Moreover, it has many applications in several fields of computational physics. We first derive a matrix method to solve linear problems. In this method, the given linear problem is reduced to a linear system of algebraic equations. The detailed convergence analysis of the proposed matrix method is given. An iterative matrix method is then constructed for nonlinear problems. The nonlinear problem is first replaced with a sequence of linear problems by utilizing the quasilinearization technique. Then, this sequence of problems is successively solved using the matrix method. Numerical examples are included to demonstrate the efficiency and accuracy of the proposed methods.

Singular Sturm–Liouville operators with extreme properties that generate black holes

AbstractThis study is motivated by several engineering papers that describe the so‐called acoustic black hole; for example, a beam with a monotonically decreasing thickness toward one of the endpoints. It appears that the time of propagation of a signal toward such an endpoint is infinite so that the signal is “trapped.” Also, the amplitude of a signal increases with no bound near this point. The main objective of this paper is the rigorous study of the spectral properties of the Sturm–Liouville problem for a second‐order differential operator on a finite interval with the coefficients vanishing at one of the endpoints—that is, with extreme coefficients. Physically, that means that we study a rod instead of a beam. We classify the endpoints, compute the essential spectrum, and determine conditions for the absence of positive eigenvalues of the corresponding self‐adjoint extensions. The absence of positive discrete spectrum means, physically, that the rod does not sound on any frequency. Our analysis allows us to precisely describe how the coefficients of the differential operator should vanish to produce the essential spectrum and where it is located. An extensive mathematical literature is devoted to the three‐dimensional problems of elasticity in bounded domains with a cusp and similar problems for the bodies with a blunted pick. The engineering papers also study the physical properties of an almost sharp beam; that is, the thickness is decreasing toward a small positive limit. In our paper, the spectral properties of the Sturm–Liouville problem with the coefficients almost vanishing at one of the endpoints are studied. We also show that, for large values of the spectral parameter, the approximation to the solution satisfies the properties found by engineers in their models; that is, the time of propagation toward the sharp end is infinite and the amplitude near that end increases with no bound.

A unified construction of product formulas and convolutions for Sturm–Liouville operators

We establish a positive product formula for the solutions of the Sturm–Liouville equation $$\ell (u) = \lambda u$$ , where $$\ell $$ belongs to a general class which includes singular and degenerate Sturm–Liouville operators. Our technique relies on a positivity theorem for possibly degenerate hyperbolic Cauchy problems and on a regularization method which makes use of the properties of the diffusion semigroup generated by the Sturm–Liouville operator. We show that the product formula gives rise to a probability preserving convolution algebra structure on the space of finite measures which satisfies the basic axioms for developing harmonic analysis on the convolution algebra. Unlike previous works, our framework includes a subfamily of Sturm–Liouville operators for which the support of the convolution of Dirac measures is noncompact. The connection with hypergroup theory is discussed. Convolution-type integral equations on weighted Lebesgue spaces are also discussed, and a solvability condition is established.

A universality law for sign correlations of eigenfunctions of differential operators

We establish a sign correlation universality law for sequences of functions \{w_n\}_{n \in \mathbb{N}} satisfying a trigonometric asymptotic law. Our results are inspired by the classical WKB asymptotic approximation for Sturm–Liouville operators, and in particular we obtain non-trivial sign correlations for eigenfunctions of generic Schrödinger operators (including the harmonic oscillator), as well as Laguerre and Chebyshev polynomials. Given two distinct points x, y \in \mathbb{R} , we ask how often do w_n(x) and w_n(y) have the same sign. Asymptotically, one would expect this to be true half the time, but this turns out to not always be the case. Under certain natural assumptions, we prove that, for all x \neq y , \frac{1}{3} \leq \lim_{N \to \infty}{ \frac{1}{N} \# \{0 \leq n < N\colon \mathrm {sgn}(w_n(x)) = \mathrm {sgn}(w_n(y)) \}} \leq \frac{2}{3}, and that these bounds are optimal, and can be attained. Our methods extend to other problems of similar flavor and we also discuss a number of open problems.

Quantum graphs on radially symmetric antitrees

We investigate spectral properties of Kirchhoff Laplacians on radially symmetric antitrees. This class of metric graphs admits a lot of symmetries, which enables us to obtain a decomposition of the corresponding Laplacian into the orthogonal sum of Sturm–Liouville operators. In contrast to the case of radially symmetric trees, the deficiency indices of the Laplacian defined on the minimal domain are at most one and they are equal to one exactly when the corresponding metric antitree has finite total volume. In this case, we provide an explicit description of all self-adjoint extensions including the Friedrichs extension. Furthermore, using the spectral theory of Krein strings, we perform a thorough spectral analysis of this model. In particular, we obtain discreteness and trace class criteria, a criterion for theKirchhoff Laplacian to be uniformly positive and provide spectral gap estimates. We show that the absolutely continuous spectrumis in a certain sense a rare event, however, we also present several classes of antitrees such that the absolutely continuous spectrum of the corresponding Laplacian is [O, \infty) .

On the Expansion Formula for a Class of Sturm–Liouville Operators

On the Expansion Formula for a Class of Sturm–Liouville Operators