Singular Sturm-Liouville Problems Research Articles

A mathematical description of the Weber nucleus as a classical and quantum mechanical system

Wilhelm Weber’s electrodynamics is an action-at-a-distance theory which has the property that equal charges inside a critical radius become attractive. Weber’s electrodynamics inside the critical radius can be interpreted as a classical Hamiltonian system whose kinetic energy is, however, expressed with respect to a Lorentzian metric. In this article we study the Schrödinger equation associated with this Hamiltonian system, and relate it to Weyl’s theory of singular Sturm–Liouville problems.

Algorithmic implementation of an exact three-point difference scheme for a certain class of singular Sturm–Liouville problems

In this article, we present a new algorithmic implementation of exact three-point difference schemes for a certain class of singular Sturm–Liouville problems. We demonstrate that computing the coefficients of the exact scheme at any grid node xj requires solving two auxiliary Cauchy problems for the second-order linear ordinary differential equations: one problem on the interval [xj−1,xj] (forward) and one problem on the interval [xj,xj+1] (backward). We have also proven the coefficient stability theorem for the exact three-point difference scheme.

Algorithmic realization of exact three‐point difference scheme for singular Sturm–Liouville problem

AbstractA new algorithmic realization of exact three‐point difference schemes for a class of singular Sturm–Liouville problems is presented. It is shown that to compute the coefficients of the exact scheme in an arbitrary grid node, it is necessary to solve two auxiliary initial value problems for the second order linear ordinary differential equations: one problem on the interval (forward) and one problem on the interval (backward). Finally, the coefficient stability of the exact three‐point difference scheme is proved.

Using the Accelerated Convergence Method for Solving the Singular Sturm–Liouville Problem

This article is dedicated to the memory of L.D. Akulenko, with whom the author of the article worked for more than 40 years. Within the framework of the accelerated convergence method developed jointly, a number of classes of problems related to the Sturm–Liouville problems were solved. Based on the research results, several dozen articles and a generalizing monograph [1] were published. In this paper, we describe the adaptation of the method to solving singular Sturm–Liouville problems.

Eigenvalues of the Laplace-Beltrami operator on a prolate spheroid

The Laplace-Beltrami operator on a prolate spheroid admits a sequence of eigenvalues. These eigenvalues are determined by a singular Sturm-Liouville problem. Properties of the eigenvalues are obtained using the minimum-maximum principle and the Prüfer angle. In particular, eigenvalues are approximated by those of generalized matrix eigenvalue problems, and their behavior is studied when the eccentricity of the spheroid approaches 0 or 1.

A Class of Singular Sturm–Liouville Problems with Discontinuity and an Eigenparameter-Dependent Boundary Condition

In this paper, we study a singular Sturm–Liouville problem with an eigenparameter-dependent boundary condition and transmission conditions at two interior points. Using an operator-theoretical formulation, we transfer the problem to an operator in an appropriate Hilbert space. It is proved that the operator is self-adjoint. We also give the asymptotic formulas of the eigenvalues of the problem. Moreover, Green’s function is also discussed.

An O(1) algorithm for the numerical evaluation of the Sturm-Liouville eigenvalues of the spheroidal wave functions of order zero

In addition to being the eigenfunctions of the restricted Fourier operator, the angular spheroidal wave functions of the first kind of order zero and nonnegative integer characteristic exponents are the solutions of a singular self-adjoint Sturm-Liouville problem. The running time of the standard algorithm for the numerical evaluation of their Sturm-Liouville eigenvalues grows with both bandlimit and characteristic exponent. Here, we describe a new approach whose running time is bounded independent of these parameters. Although the Sturm-Liouville eigenvalues are of little interest themselves, our algorithm is a component of a fast scheme for the numerical evaluation of the prolate spheroidal wave functions developed by one of the authors. We illustrate the performance of our method with numerical experiments.

A singular Sturm–Liouville problem with limit circle endpoint and boundary conditions rationally dependent on the eigenparameter

A singular Sturm–Liouville problem with limit circle endpoint and boundary conditions rationally dependent on the eigenparameter

Research on singular Sturm–Liouville spectral problems with a weighted function

As early as 1910, Weyl gave a classification of the singular Sturm–Liouville equation, and divided it into the Limit Point Case and the Limit Circle Case at infinity. This led to the study of singular Sturm–Liouville spectrum theory. With the development of applications, the importance of singular Sturm–Liouville problems with a weighted function becomes more and more significant. This paper focuses on the study of singular Sturm–Liouville problems with a weighted function. Finally, an example of singular Sturm–Liouville problems with a weighted function is given.

Characterization of symmetric operators and their Friedrichs extension for singular Sturm-Liouville problems

We find, explicitly, the boundary conditions which determine the symmetric extensions of the minimal operator. For each of these operators which is bounded below we find a boundary condition which determines its Friedrichs extension.

Dynamic identification of pretensile forces in a spider orb-web

In this paper we propose a constructive procedure to determine the radial pretension in an axially-symmetric orb-web, on the basis of one eigenfrequency and its corresponding vibration mode. The method can be applied both to axisymmetric modes, corresponding to the solution of a regular Sturm–Liouville problem, and to non-axisymmetric modes, corresponding to the solution of a singular Sturm–Liouville problem. In the absence of measurement errors of the eigenfunction, the identification of the pretension field is almost perfect. On the contrary, if the measurement presents some noise, then identification is imprecise. In this case, a regularization technique applied to the measurement of the eigenfunction, or to its derivative, is proposed, as well as a filtering process, which significantly improves the reconstruction of the pretension field. The fundamental mode of vibration, sampled with a suitable number of points, is the one that provides the lowest identification error.

Solutions of Sturm-Liouville Problems

This paper further improves the Lie group method with Magnus expansion proposed in a previous paper by the authors, to solve some types of direct singular Sturm–Liouville problems. Next, a concrete implementation to the inverse Sturm–Liouville problem algorithm proposed by Barcilon (1974) is provided. Furthermore, computational feasibility and applicability of this algorithm to solve inverse Sturm–Liouville problems of higher order (for n=2,4) are verified successfully. It is observed that the method is successful even in the presence of significant noise, provided that the assumptions of the algorithm are satisfied. In conclusion, this work provides a method that can be adapted successfully for solving a direct (regular/singular) or inverse Sturm–Liouville problem (SLP) of an arbitrary order with arbitrary boundary conditions.

Analytical Approaches to the Analysis of Unsteady Heat Conduction for Partially Bounded Regions

A mathematical theory is developed for the construction of integral transforms for the following partially bounded regions: a space with an internal cylindrical cavity in the cylindrical coordinates (radial heat flux); a space with an internal spherical cavity in the spherical coordinates (central symmetry); and a space bounded by a planar surface in the Cartesian coordinates. Expressions are proposed for the integral transforms, Laplace operator images, and inversions for images. The formulated approach differs from the classical theory of differential equations of mathematical physics for the construction of integral transforms with a continuous spectrum of eigenvalues based on the corresponding singular Sturm–Liouville problems. The proposed method is based on the operational solutions of the initial boundary-value problems of unsteady heat conduction with an inhomogeneous initial function and homogeneous boundary conditions. The formulated approach makes it possible to develop the Green’s function method and to construct integral representations of the analytical solutions of the boundary-value problems simultaneously based on the Green’s functions and inhomogeneities in the main equation and boundary conditions of the problem. The proposed functional relations can be used in numerous particular cases of practical thermal physics. Examples of the application of the obtained results in some fields of science and technology are presented.

Dispersive Landau levels and valley currents in strained graphene nanoribbons

We describe a simple setup generating pure valley currents -- valley transport without charge transport -- in strained graphene nanoribbons with zigzag edges. The crucial ingredient is a uniaxial strain pattern which couples to the low-energy Dirac electrons as a uniform pseudomagnetic field. Remarkably, the resulting pseudo-Landau levels are not flat but disperse linearly from the Dirac points, with an opposite slope in the two valleys. We show how this is a natural consequence of an inhomogeneous Fermi velocity arising in the low-energy theory describing the system, which maps to an exactly-solvable singular Sturm-Liouville problem. The velocity of the valley currents can be controlled by tuning the magnitude of strain and by applying bias voltages across the ribbon. Furthermore, applying an electric field along the ribbon leads to pumping of charge carriers between the two valleys, realizing a valley analog of the chiral anomaly in one spatial dimension. These effects produce unique signatures that can be observed experimentally by performing ordinary charge transport measurements and spectroscopy.

Asymptotic Behavior for Eigenvalues of the Singular Sturm-Liouville Problem with Transmission Conditions

In this study, we consider the singular Sturm-Liouville (S-L) problem with transmission conditions (TC). The eigenvalues of the singular S-L problem with TC are investigated. By defining a new Hilbert space which is related to TC, the self-adjointness of the S-L problem in this associated Hilbert space is proved. And we give the condition for $$\lambda $$ being the eigenvalue of the singular S-L problem. Furthermore, the asymptotic behavior of eigenvalues of the singular S-L problem is described.

A corrected spectral method for Sturm–Liouville problems with unbounded potential at one endpoint

In this paper, we shall derive a spectral matrix method for the approximation of the eigenvalues of (weakly) regular and singular Sturm–Liouville problems in normal form with an unbounded potential at the left endpoint. The method is obtained by using a Galerkin approach with an approximation of the eigenfunctions given by suitable combinations of Legendre polynomials. We will study the errors in the eigenvalue estimates for problems with unsmooth eigenfunctions in proximity of the left endpoint. The results of this analysis will be then used conveniently to determine low-cost and effective procedures for the computation of corrected numerical eigenvalues. Finally, we shall present and discuss the results of several numerical experiments which confirm the effectiveness of the approach.

Computing High-Index Eigenvalues of Singular Sturm–Liouville Problems

This paper deals with the computation of the high-index eigenvalues of singular Sturm–Liouville problems using the Chebyshev spectral collocation method. The singular Sturm–Liouville problem is transformed into generalized eigenvalue problem by using the spectral differentiation matrices to compute derivatives of Chebyshev polynomials at Chebyshev Gauss–Lobatto nodes. A few different examples shall be solved numerically to demonstrate reliability and efficiency of the proposed technique.

On Spectral Approximations with Nonstandard Weight Functions and Their Implementations to Generalized Chaos Expansions

In this manuscript, we analyze the expansions of functions in orthogonal polynomials associated with a general weight function in a multidimensional setting. Such orthogonal polynomials can be obtained, e.g, by Gram–Schmidt orthogonalization. However, in most cases, they are not eigenfunctions of some singular Sturm–Liouville problem, as is the case for known polynomials, such as the Jacobi polynomials. Therefore, the standard convergence theorems do not apply. Furthermore, since in general multidimensional cases the weight functions are not a tensor product of one-dimensional functions, the orthogonal polynomials are not a product of one-dimensional orthogonal polynomials, as well. This work provides a way of estimating the convergence rate using a comparison lemma. We also present a spectrally convergent, multidimensional, integration method. Numerical examples demonstrate the efficacy of the proposed method. We also show that the use of non-standard weight functions can allow for efficient integration of singular functions. We demonstrate the use of this method to uncertainty quantification problem using Generalized Polynomial Chaos Expansions in the case of dependent random variables, as well.

INTEGRAL TRANSFORMATION IN A PARTIALLY BOUNDED REGION WITH A RADIAL THERMAL FLOW

A mathematical theory is developed for constructing integral transformations in a partially bounded region with a radial heat flow - a massive body bounded from the inside by a cylindrical cavity. Constructed: an integral transformation, the image of the operator on the right side of the equation of unsteady heat conduction, the inversion formula for the image of the desired function. The proposed approach favorably differs from the classical theory of differential equations of mathematical physics for the construction of generalized integral transformations based on the eigenfunctions of the corresponding singular Sturm-Liouville problems. The developed method is based on the operational solution of the initial boundary problems of unsteady heat conduction with an initial function of a general form L2(r0,∞) belonging to the r > r0 region and homogeneous boundary conditions and is associated with the calculation of the Riemann-Mellin contour integrals from images containing various combinations of modified Bessel functions. At the same time, for the above-mentioned region, the method of Green's functions was developed by constructing integral representations of analytical solutions of the first, second and third boundary value problems through inhomogeneities in the initial formulation of the problem (boundary conditions, source function in the initial equation). Mathematical models for finding the corresponding Green's functions are formulated, and functional relations of all three Green functions included in the presented integral formula are written out with the help of the developed theory of integral transformations. The functional relations constructed in the article can be used when considering numerous special cases of practical thermal physics. The specific possible applications of the presented results in many areas of science and technology are given.

From parabolic to loxodromic BMS transformations

Half of the Bondi–Metzner–Sachs (BMS) transformations consist of orientation-preserving conformal homeomorphisms of the extended complex plane known as fractional linear (or Möbius) transformations. These can be of 4 kinds, i.e. they are classified as being parabolic, or hyperbolic, or elliptic, or loxodromic, depending on the number of fixed points and on the value of the trace of the associated $$2 \times 2$$ matrix in the projective version of the $$SL(2,\mathbb {C})$$ group. The resulting particular forms of $$SL(2,\mathbb {C})$$ matrices affect also the other half of BMS transformations, and are used here to propose 4 realizations of the asymptotic symmetry group that we call, again, parabolic, or hyperbolic, or elliptic, or loxodromic. In the second part of the paper, we prove that a subset of hyperbolic and loxodromic transformations, those having trace that approaches $$\infty $$ , correspond to the fulfillment of limit-point condition for singular Sturm–Liouville problems. Thus, a profound link may exist between the language for describing asymptotically flat space-times and the world of complex analysis and self-adjoint problems in ordinary quantum mechanics.