Sturm-Liouville Research Articles

Inverse nodal problems for singular diffusion equation

In this study, some properties of the pencils of singular Sturm–Liouville operators are investigated. Firstly, the behaviors of eigenvalues and eigenfunctions is learned, then for each discontinuity point a solution of the inverse problem is given to determine the potential function and parameters , and with the help of a dense set of nodes. And finally, a constructive method is given for solving the given inverse problem.

Variational approach to a symmetric boundary value problem generated by a system of equations and separated boundary conditions

This work provides some information on the eigenvalues and eigenfunctions of a problem which is constructed by a system of equations and symmetric boundary conditions that includes the ordinary second‐order Sturm–Liouville boundary value problem. In particular, we show that the problem has an infinite number of discrete eigenvalues with a greatest lower bound and the corresponding eigenfunctions are complete in mean and energy. We introduce the results using the variational approach that enables us to consider only continuous pair functions instead of absolutely continuous pair functions.

Study of Nonlinear Second-Order Differential Inclusion Driven by a Φ−Laplacian Operator Using the Lower and Upper Solutions Method

In this paper, we study a second-order differential inclusion under boundary conditions governed by maximal monotone multivalued operators. These boundary conditions incorporate the classical Dirichlet, Neumann, and Sturm–Liouville problems. Our method of study combines the method of lower and upper solutions, the analysis of multivalued functions, and the theory of monotone operators. We show the existence of solutions when the lower solution σ and the upper solution γ are well ordered. Next, we show how our arguments of proof can be easily exploited to establish the existence of extremal solutions in the functional interval σ,γ. We also show that our method can be applied to the periodic case.

Uniqueness of Nonlinear Inverse Problem for Sturm–Liouville Operator with Multiple Delays

The inverse problem concerns how to reconstruct the operator from given spectral data. The main goal of this paper is to address nonlinear inverse Sturm–Liouville problem with multiple delays. By using a new technique and method: zero function extension, we establish the uniqueness result and practical method for recovering the nonlinear inverse problem from two spectra.

A REPRESENTATION FORMULA FOR THE RESOLVENT OF CONFORMABLE FRACTIONAL STURM-LIOUVILLE OPERATOR

In this study, the resolvent of the conformable fractional Sturm–Liouville operator is considered. An integral representation for the resolvent of this operator is obtained.

Optimal inverse problems of potentials for two given eigenvalues of Sturm–Liouville problems

The present paper is concerned with the infimum of the norm of potentials for Sturm–Liouville eigenvalue problems with Dirichlet boundary condition such that the first two eigenvalues are known. The explicit quantity of the infimum is given by the two eigenvalues.

Existence and Uniqueness of Some Unconventional Fractional Sturm–Liouville Equations

In this paper, we provide existence and uniqueness results for the initial value problems associated with mixed Riemann–Liouville/Caputo differential equations in the real domain. We show that, under appropriate conditions in a fractional order, solutions are always square-integrable on the finite interval under consideration. The results are valid for equations that have sign-indefinite leading terms and measurable coefficients. Existence and uniqueness theorem results are also provided for two-point boundary value problems in a closed interval.

ON THE UNIQUENESS OF INVERSE PROBLEMS FOR ”WEIGHTED” STURM-LIOUVILLE OPERATOR WITH DELTA INTERACTION

We establish various uniqueness result of inverse problems for ”weighted” Sturm-Liouville operator with interaction point.

Reconstruction techniques for complex potentials

An approach for solving a variety of inverse coefficient problems for the Sturm–Liouville equation −y″ + q(x)y = ρ2y with a complex valued potential q(x) is presented. It is based on Neumann series of Bessel functions representations for solutions. With their aid the problem is reduced to a system of linear algebraic equations for the coefficients of the representations. The potential is recovered from an arithmetic combination of the first two coefficients. Special cases of the considered problems include the recovery of the potential from a Weyl function, inverse two-spectrum Sturm–Liouville problems, as well as the inverse scattering problem on a finite interval. The approach leads to efficient numerical algorithms for solving coefficient inverse problems. Numerical efficiency is illustrated by several examples.

Sturm–Liouville Problem for a One-Dimensional Thermoelastic Operator in Cartesian, Cylindrical, and Spherical Coordinate Systems

Sturm–Liouville Problem for a One-Dimensional Thermoelastic Operator in Cartesian, Cylindrical, and Spherical Coordinate Systems

Study of the Asymptotic Properties of the Solution to a Problem with a Parameter for the Sturm–Liouville Operator with a Singular Potential

Study of the Asymptotic Properties of the Solution to a Problem with a Parameter for the Sturm–Liouville Operator with a Singular Potential

THE REGULARIZED TRACE FORMULA FOR A DISCONTINUOUS STURM-LIOUVILLE PROBLEM WITH A SPECTRAL PARAMETER DEPENDENT JUMP CONDITION

The first regularized trace formula for a discontinuous Sturm-Liouville problem with a spectral parameter dependent jump condition is obtained.

INVERSE NODAL PROBLEMS FOR PENCILS OF SINGULAR STURM-LIOUVILLE OPERATORS

In this study, some properties of the pencils of singular Sturm-Liouville operators are investigated. Firstly, the behaviors of eigenvalues and eigenfunctions is learned, then for each discontinuity point

Solvability of the inverse spectral problem for Sturm-Liouville operator with spectral parameter in the boundary condition

This paper deals with an inverse problem for the Sturm-Liouville operator with non-separated boundary conditions, one of which linearly depends on a spectral parameter. Uniqueness theorem is proved, solution algorithm is constructed and sufficient conditions for solvability of inverse problem are obtained. As spectral data, we only use the spectrum of one boundary value problem and some sequence of signs, and some number.

ON AN INVERSE SCATTERING PROBLEM FOR THE QUADRATIC PENCIL OF THE STURM-LIOUVILLE OPERATOR WITH A PIECE-WISE CONSTANT COEFFICIENT

In this work, direct and inverse scattering problems on the real axis for the quadratic pencil of the Sturm-Liouville operator with piece-wise constant coefficient are studied. The new integral representations for solutions are given, the scattering data is defined, the main integral equations of the inverse scattering problem are obtained, the spectral characteristics of the scattering data are investigated and uniqueness theorem for the solution of inverse problem is proven, the necessary and sufficient conditions for recovering of the potentials are examined.

Uniqueness of Finite Exceptional Orthogonal Polynomial Sequences Composed of Wronskian Transforms of Romanovski-Routh Polynomials

This paper exploits two remarkable features of the translationally form-invariant (TFI) canonical Sturm–Liouville equation (CSLE) transfigured by Liouville transformation into the Schrödinger equation with the shape-invariant Gendenshtein (Scarf II) potential. First, the Darboux–Crum net of rationally extended Gendenshtein potentials can be specified by a single series of Maya diagrams. Second, the exponent differences for the poles of the CSLE in the finite plane are energy-independent. The cornerstone of the presented analysis is the reformulation of the conventional supersymmetric (SUSY) quantum mechanics of exactly solvable rational potentials in terms of ‘generalized Darboux transformations’ of canonical Sturm–Liouville equations introduced by Rudyak and Zakhariev at the end of the last century. It has been proven by the author that the first feature assures that all the eigenfunctions of the TFI CSLE are expressible in terms of Wronskians of seed solutions of the same type, while the second feature makes it possible to represent each of the mentioned Wronskians as a weighted Wronskian of Routh polynomials. It is shown that the numerators of the polynomial fractions in question form the exceptional orthogonal polynomial (EOP) sequences composed of Wronskian transforms of the given finite set of Romanovski–Routh polynomials excluding their juxtaposed pairs, which have already been used as seed polynomials.

Reconstruction techniques for quantum trees

The inverse problem of recovery of a potential on a quantum tree graph from the Weyl matrix given at a number of points is considered. A method for its numerical solution is proposed. The overall approach is based on the leaf peeling method combined with Neumann series of Bessel functions (NSBF) representations for solutions of Sturm–Liouville equations. In each step, the solution of the arising inverse problems reduces to dealing with the NSBF coefficients. The leaf peeling method allows one to localize the general inverse problem to local problems on sheaves, while the approach based on the NSBF representations leads to splitting the local problems into two‐spectrum inverse problems on separate edges and reduces them to systems of linear algebraic equations for the NSBF coefficients. Moreover, the potential on each edge is recovered from the very first NSBF coefficient. The proposed method leads to an efficient numerical algorithm that is illustrated by numerical tests.

Higher-dimensional holographic superconductors in Born–Infeld electrodynamics and f(R) gravity

In this paper, the properties of higher dimensional holographic superconductors are studied in the background of f(R) gravity and Born–Infeld electrodynamics. A specific model of f(R) gravity is considered, allowing a perturbative approach to the problem. The Sturm–Liouville eigenvalue problem is used to analytically calculate the critical temperature and the condensation operator. An expression for the critical temperature in terms of the charge density including the correction from modified gravity is derived. It is seen that the higher values of the Born–Infeld coupling parameter make the condensation harder to form. In addition, the limiting values of this parameter, above which Born–Infeld electrodynamics cannot be applied, are found for different dimensions. Another interesting property is that the increasing modifications of f(R) gravity lead to larger values of the critical temperature and a decrease in the condensation gap, which means that the condensation is easier to form.

Discrete Sturm-Liouville equations with point interaction

Discrete Sturm-Liouville equations with point interaction

Spectral subspaces of Sturm-Liouville operators and variable bandwidth

We study spectral subspaces of the Sturm-Liouville operator f↦−(pf′)′ on R, where p is a positive, piecewise constant function. Functions in these subspaces can be thought of as having a local bandwidth determined by 1/p. Using the spectral theory of Sturm-Liouville operators, we make the reproducing kernel of these spectral subspaces more explicit and compute it completely in certain cases. As a contribution to sampling theory, we then prove necessary density conditions for sampling and interpolation in these subspaces and determine the critical density that separates sets of stable sampling from sets of interpolation.