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.     

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.     

On sharp asymptotic formulas for the Sturm–Liouville operator with a matrix potential

In this article we obtain the sharp asymptotic formulas for the eigenvalues and eigenfunctions of the non-self-adjoint operators generated by a system of the Sturm–Liouville equations with Dirichlet and Neumann boundary conditions. Using these asymptotic formulas, we find a condition on the potential for which the root functions of these operators form a Riesz basis.     

Reconstruction of Energy-Dependent Sturm–Liouville Equations from two Spectra

In this paper we study the inverse spectral problem of reconstructing energy-dependent Sturm–Liouville equations from two spectra. We give a reconstruction algorithm and establish existence and uniqueness of reconstruction. Our approach essentially exploits the connection between the spectral problems under study and those for Dirac operators of a special form.     

On the deficiency index of the vector-valued Sturm–Liouville operator

Let R₊:= [0, +∞), and let the matrix functions P, Q, and R of order n, n ∈ N, defined on the semiaxis R₊ be such that P(x) is a nondegenerate matrix, P(x) and Q(x) are Hermitian matrices for x ∈ R₊ and the elements of the matrix functions P^?1, Q, and R are measurable on R₊ and summable on each of its closed finite subintervals. We study the operators generated in the space L_n²(R₊) by formal expressions of the form l[f] = ?(P(f' ? Rf))' ? R*P(f' ? Rf) + Qf and, as a particular case, operators generated by expressions of the form l[f] = ?(P₀f')' + i((Q₀f)' + Q₀f') + P'₁f, where everywhere the derivatives are understood in the sense of distributions and P₀, Q₀, and P₁ are Hermitianmatrix functions of order n with Lebesgue measurable elements such that P₀^?1 exists and ∥P0∥, ∥P₀^?1∥, ∥P₀^?1∥∥P₁∥², ∥P₀^?1∥∥Q₀∥² ∈ L_loc¹(R₊). Themain goal in this paper is to study of the deficiency index of the minimal operator L₀ generated by expression l[f] in L_n²(R₊) in terms of the matrix functions P, Q, and R (P₀, Q₀, and P₁). The obtained results are applied to differential operators generated by expressions of the form \(l[f] = - f'' + \sum\limits_{k = 1}^{ + \infty } {{H_k}} \delta \left( {x - {x_k}} \right)f\), where x_k, k = 1, 2,..., is an increasing sequence of positive numbers, with lim_k→+∞x_k = +∞, H_k is a number Hermitian matrix of order n, and δ(x) is the Dirac δ-function.     

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.     

On the reconstruction of the potential in the inverse problem for a power of the Sturm—Liouville operator

In this paper, we prove the existence and uniqueness theorem for the inverse problem of spectral analysis for a power of the Sturm—Liouville operator on an interval with power > 3/4 and a quadratically summable, complex-valued, nonsymmetric potential. In this case, the potential is reconstructed by the mix of four spectra or by two spectra.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 8 , Suzdal Conference—2, 2003.This revised version was published online in April 2005 with a corrected cover date.     

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.     

Irregular boundary value problem for the Sturm–Liouville operator

We consider an eigenvalue problem for the Sturm–Liouville operator with nonclassical asymptotics of the spectrum. We prove that this problem, which has a complete system of root functions, is not almost regular (Stone-regular) but its Green function has a polynomial order of growth in the spectral parameter.     

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.     

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.     

Distribution of the spectrum of a singular positive Sturm–Liouville operator perturbed by the Dirac delta function

We consider the Sturm–Liouville operator generated in the space L ²[0,+∞) by the expression l _a,b:= ?d ²/dx ² +x+aδ(x?b) and the boundary condition y(0) = 0. We prove that the eigenvalues λ _n of this operator satisfy the inequalities λ₁ ⁰ < λ₁ < λ₂ ⁰ and λ_n ⁰ ≤ λ_n < λ_n+1 ⁰, n = 2, 3,..., where {?λ_n ⁰} is the sequence of zeros of the Airy function Ai (λ). We find the asymptotics of λ_n as n → +∞ depending on the parameters a and b.     

On the determination of the impulsive Sturm–Liouville operator with the eigenparameter-dependent boundary conditions

In the present work, we consider the inverse problem for the impulsive Sturm–Liouville equations with eigenparameter-dependent boundary conditions on the whole interval (0,π) from interior spectral data. We prove two uniqueness theorems on the potential q(x) and boundary conditions for the interior inverse problem, and using the Weyl function technique, we show that if coefficients of the first boundary condition, that is, h₁,h₂, are known, then the potential function q(x) and coefficients of the second boundary condition, that is, H₁,H₂, are uniquely determined by information about the eigenfunctions at the midpoint of the interval and one spectrum or partial information on the eigenfunctions at some internal points and some of two spectra.     

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 existence of infinitely many eigenvalues in a nonlinear Sturm–Liouville problem arising in the theory of waveguides

We consider a nonlinear eigenvalue problem of the Sturm–Liouville type on an interval with boundary conditions of the first kind. The problem describes the propagation of polarized electromagnetic waves in a plane two-layer dielectric waveguide. The cases of a homogeneous and an inhomogeneous medium are studied. The existence of infinitely many positive and negative eigenvalues is proved.     

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...