Non-real eigenvalues of indefinite Sturm–Liouville problems

The present paper deals with non-real eigenvalues of regular indefinite Sturm–Liouville problems. A priori bounds and sufficient conditions of the existence for non-real eigenvalues are obtained under mild integrable conditions of coefficients.     

Numerical computation of eigenvalues of discontinuous Sturm–Liouville problems with parameter dependent boundary conditions using sinc method

In this paper, we consider a Sturm–Liouville problem which contains an eigenparameter appearing linearly in two boundary conditions, in addition to an internal point of discontinuity. Eigenvalue problems with eigenparameter appearing in the boundary conditions usually have complicated characteristic determinant where zeros cannot be explicitly computed. We apply the sinc method, which is based on the sampling theory to compute approximations of the eigenvalues. An error analysis is exhibited involving rigorous error bounds. Using computable error bounds we obtain eigenvalue enclosures in a simple way. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

Computation of eigenvalues and solutions of regular Sturm–Liouville problems using Haar wavelets

The paper presents a novel method for the computation of eigenvalues and solutions of Sturm–Liouville eigenvalue problems (SLEPs) using truncated Haar wavelet series. This is an extension of the technique proposed by Hsiao to solve discretized version of variational problems via Haar wavelets. The proposed method aims to cover a wider class of problems, by applying it to historically important and a very useful class of boundary value problems, thereby enhancing its applicability. To demonstrate the effectiveness and efficiency of the method various celebrated Sturm–Liouville problems are analyzed for their eigenvalues and solutions. Also, eigensystems are investigated for their asymptotic and oscillatory behavior. The proposed scheme, unlike the conventional numerical schemes, such as Rayleigh quotient and Rayleigh–Ritz approximation, gives eigenpairs simultaneously and provides upper and lower estimates of the smallest eigenvalue, and it is found to have quadratic convergence with increase in resolution.     

Computing eigenvalues of Sturm–Liouville problems by Hermite interpolations

We establish a new method to compute the eigenvalues of Sturm?CLiouville problems by the use of Hermite interpolations at equidistant nodes. We rigorously give estimates for the error by considering both truncation and amplitude errors. We compare the results of the new technique with those involving the classical sinc method as well as a SLEIGN2-based method. We also introduce curves that illustrate the enclosure intervals.     

Error estimate of eigenvalues of perturbed second-order discrete Sturm–Liouville problems

This paper is concerned with eigenvalues of perturbed second-order vector discrete Sturm–Liouville problems. By some variational properties of eigenvalues of discrete Sturm–Liouville problems, error estimates of eigenvalues of perturbed problems, sufficiently close to a given Sturm–Liouville problem, are given under a certain non-singularity condition. Perturbations of the coefficient functions of the difference equation, the weight function, and the coefficients of the boundary condition are all considered. This, together with higher-dimension involved, results in a certain complexity of the problem and difficulty of study. As a direct consequence, continuous dependence of eigenvalues on problems is obtained under the non-singularity condition. In addition, an example is presented to illustrate the necessity of the non-singularity condition.     

Inequalities among eigenvalues of Sturm–Liouville problems with distribution potentials

This paper deals with the eigenvalue problems for the Sturm–Liouville operators generated by the differential expression

L(y)=−(p(x)y^′)^′+q(x)y$L\left(y\right)=-{\left(p\left(x\right)y^{\prime }\right)}^{\prime }+q\left(x\right)y$

with singular coefficients q(x)$q\left(x\right)$ in the sense of distributions. We obtain the inequalities among the eigenvalues corresponding to different self-adjoint boundary conditions. The continuity region, the differentiability and the monotonicity of the n$n$th eigenvalue corresponding to the separated boundary conditions are given. Oscillation properties of the eigenfunctions of all the coupled Sturm–Liouville problems are characterized. The main results of this paper can also be applied to solve a class of transmission problems.     

On the multiplicity of eigenvalues of the Sturm–Liouville problem on graphs

Bounds for the multiplicity of the eigenvalues of the Sturm–Liouville problem on a graph, which are valid for a wide class of consistency (transmission) conditions at the vertices of the graph, are given. The multiplicities are estimated using the topological characteristics of the graph. In the framework of the notions that we use, the bounds turn out to be exact.     

Distribution of eigenvalues and trace formula for the Sturm–Liouville operator equation

We study the asymptotic distribution of eigenvalues of the problem generated by the Sturm–Liouville operator equation. A formula for the regularized trace of the corresponding operator is obtained.     

An efficient technique for finding the eigenvalues of fourth-order Sturm–Liouville problems

In this paper, we will develop a numerical technique for finding the eigenvalues of fourth-order non-singular Sturm–Liouville problems. We used the variational iteration methods as a basis for this technique. Numerical results and conclusions will be presented. Comparison results with others will be presented.     

Chebyshev differentiation matrices for efficient computation of the eigenvalues of fourth-order Sturm–Liouville problems

In this paper, an efficient technique based on the Chebyshev spectral collocation method for computing the eigenvalues of fourth-order Sturm–Liouville boundary value problems is proposed. The excellent performance of this scheme is illustrated through four examples. Numerical results and comparison with other methods are presented.     

Two very accurate and efficient methods for computing eigenvalues of Sturm–Liouville problems

In this paper, we discuss and compare two useful schemes for Sturm–Liouville eigenvalue problems: differential quadrature method (DQM) and collocation method with sinc functions. These methods are then tested on a few examples and a comparison is made. It is shown that the sinc-collocation method in many instances gives better results.     

On the second-order Fréchet derivatives of eigenvalues of Sturm–Liouville problems in potentials

Archiv der Mathematik - The works of V.A. Vinokurov have shown that eigenvalues and normalized eigenfunctions of Sturm–Liouville problems are analytic in potentials, considered as mappings...     

Eigenvalues of limit-point Sturm–Liouville problems

The dependence of the eigenvalues of self-adjoint Sturm–Liouville problems on the boundary conditions when each endpoint is regular or in the limit-circle case is now, due to some surprisingly recent results, well understood. Here we study this dependence for singular problems with one endpoint in the limit-point case.     

Relations among eigenvalues of left-definite Sturm–Liouville problems with coupled BCS and separated BCS

This paper deals with Left-Definite regular self-adjoint SLPs with coupled boundary conditions. It is obtained that, for a given eigenvalue _λ∗${\lambda }_{\ast }$ of Left-Definite regular SLPs with coupled BCs, there are some separated BCs also having _λ∗${\lambda }_{\ast }$ as an eigenvalue with the same index or indices. And then we can construct an algorithm for computing the index of a given eigenvalue for the Left-Definite SLPs with coupled boundary conditions.     

An inverse problem for Sturm–Liouville differential operators on A-graphs

An inverse problem of spectral analysis is studied for Sturm–Liouville differential operators on a A-graph with the standard matching conditions for internal vertices. The uniqueness theorem is proved, and a constructive solution for this class of inverse problems is obtained.     

Reflectionless Sturm–Liouville equations

We consider compactly supported perturbations of periodic Sturm–Liouville equations. In this context, one can use the Floquet solutions of the periodic background to define scattering coefficients. We prove that if the reflection coefficient is identically zero, then the operators corresponding to the periodic and perturbed equations, respectively, are unitarily equivalent. In some appendices, we also provide the proofs of several basic estimates, e.g., bounds and asymptotics for the relevant m$m$-functions.     

Infinite periodic solutions to a class of second-order Sturm–Liouville neutral differential equations

By means of variational structure and _Z2$Z_2$-group index theory, we obtain infinite periodic solutions to a class second-order Sturm–Liouville neutral delay equations

(p(t)x^′(t−sτ))^′−q(t)x(t−sτ)+f(t,x(t),x(t−τ),x(t−2τ),…,x(t−2sτ))=0.${\left(p\left(t\right)x^{\prime }(t-s\tau )\right)}^{\prime }-q\left(t\right)x(t-s\tau )+f(t,x\left(t\right),x(t-\tau ),x(t-2\tau ),\dots ,x(t-2s\tau ))=0\text{.}$