Characterization of the scattering data for the Sturm–Liouville operator

This work studies the scattering problem on the real axis for the Sturm–Liouville equation with discontinuous leading coefficient and the real‐valued steplike potential q(x) that has different constant asymptotes as x → ± ∞ . We investigate the properties of the scattering data, obtain the main integral equations of the inverse scattering problem, and also give necessary and sufficient conditions characterizing the scattering data. Copyright © 2013 John Wiley & Sons, Ltd.     

Paley–Wiener and Boas theorems for singular Sturm–Liouville integral transforms

This paper deals with a class of integral transforms arising from a singular Sturm–Liouville problem y″−q(x)y=−λy, x(a,b), in the limit-point case at one end or both ends of the interval (a,b). The paper completely solves the problem of characterization of the image of a function that has compact support (Paley–Wiener theorem) and also of a function that vanishes on some interval (Boas problem) under this class of transforms. The characterizations are obtained with no restriction on q(x) other than being locally integrable.     

Half‐inverse problems for the quadratic pencil of the Sturm–Liouville equations with impulse

In this paper, we consider the inverse spectral problem for the impulsive Sturm–Liouville differential pencils on [0, π] with the Robin boundary conditions and the jump conditions at the point . We prove that two potentials functions on the whole interval and the parameters in the boundary and jump conditions can be determined from a set of eigenvalues for two cases: (i) the potentials given on and (ii) the potentials given on , where 0 < α < 1 , respectively. Inverse spectral problems, Sturm–Liouville operator, spectrum, uniqueness.     

Multiplicity results for Sturm–Liouville boundary value problems

Multiplicity results for Sturm–Liouville boundary value problems are obtained. Proofs are based on variational methods.     

An inverse polynomial method for the identification of the leading coefficient in the Sturm–Liouville operator from boundary measurements

An inverse polynomial method of determining the unknown leading coefficient k=k(x) of the linear Sturm–Liouville operator Au=−(k(x)u^′(x))^′+q(x)u(x), x(0,1), is presented. As an additional condition only two measured data at the boundary (x=0,x=1) are used. In absence of a singular point (u^′(x)≠0,u^″(x)≠0,x[0,1]) the inverse problem is classified as a well-conditioned . If there exists at least one singular point, then the inverse problem is classified as moderately ill-conditioned (u^′(x₀)=0,x₀(0,1);u^′(x)≠0,x≠x₀;u^″(x)≠0,x[0,1]) and severely ill-conditioned (u^′(x₀)=u^″(x₀)=0,x₀(0,1);u^′(x)≠0,u^″(x)≠0,x≠x₀). For each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a Cauchy problem for to the first order differential equation with respect the unknown function k=k(x). An approximate analytical solution of the each Cauchy problems are derived in explicit form. Numerical simulations all the above cases are given for noise free and noisy data. An accuracy of the presented approach is demonstrated on numerical test solutions.     

Complete systems of recursive integrals and Taylor series for solutions of Sturm–Liouville equations

Given a regular nonvanishing complex valued solution y₀ of the equation , x ∈ (a,b), assume that it is n times differentiable at a point x₀ ∈ [a,b]. We present explicit formulas for calculating the first n derivatives at x₀ for any solution of the equation . That is, a map transforming the Taylor expansion of y₀ into the Taylor expansion of u is constructed. The result is obtained with the aid of the representation for solutions of the Sturm‐Liouville equation in terms of spectral parameter power series. Copyright © 2012 John Wiley & Sons, Ltd.     

Inverse Sturm–Liouville spectral problem on symmetric star‐tree

A spectral problem for the Sturm–Liouville equation on the edges of an equilateral regular star‐tree with the Dirichlet boundary conditions at the pendant vertices and Kirchhoff and continuity conditions at the interior vertices is considered. The potential in the Sturm–Liouville equation is a real–valued square summable function, symmetrically distributed with respect to the middle point of any edge. If {λ_j}is a sequence of real numbers, necessary and sufficient conditions for {λ_j}to be the spectrum of the problem under consideration are established. Copyright © 2013 John Wiley & Sons, Ltd.     

Linear Sturm–Liouville problems with general homogeneous linear multi‐point boundary conditions

In this paper, we study second order linear Sturm–Liouville problems involving one or two homogeneous linear multi‐point boundary conditions in the most general form. We obtain conditions for the existence of a sequence of positive eigenvalues with consecutive zero counts of the eigenfunctions. Furthermore, we reveal the interlacing relations between the eigenvalues of such Sturm–Liouville problems and those of Sturm–Liouville problems with certain two‐point separated boundary conditions.     

Inverse problems for Sturm–Liouville operators with interior discontinuities and boundary conditions dependent on the spectral parameter

In this paper, we discuss the inverse problem for Sturm–Liouville operators with arbitrary number of interior discontinuities and boundary conditions having fractional linear function of spectral parameter on the finite interval [0,1]. Using Weyl function techniques, we establish some uniqueness theorems for the Sturm–Liouville operator. Copyright © 2012 John Wiley & Sons, Ltd.     

Self‐adjoint Sturm–Liouville problems with an infinite number of boundary conditions

We study Sturm–Liouville (SL) problems on an infinite number of intervals, adjacent endpoints are linked by means of boundary conditions, and characterize the conditions which determine self‐adjoint operators in a Hilbert space which is the direct sum of the spaces for each interval. These conditions can be regular or singular, separated or coupled. Furthermore, the inner products of the summand spaces may be multiples of the usual inner products with different spaces having different multiples. We also extend the GKN Theorem to cover the infinite number of intervals theory with modified inner products and discuss the connection between our characterization and the classical one with the usual inner products. Our results include the finite number of intervals case.     

Uniqueness of solutions to inverse Sturm–Liouville problems with potential using three spectra

The uniqueness of solutions to two inverse Sturm–Liouville problems using three spectra is proven, based on the uniqueness of the solution-pair to an overdetermined Goursat–Cauchy boundary value problem. We discuss the uniqueness of the potential for a Dirichlet boundary condition at an arbitrary interior node, and for a Robin boundary condition at an arbitrary interior node, whereas at the exterior nodes we have Dirichlet boundary conditions in both situations. Here we are particularly concerned with potential functions that are L²(0,a).     

Solvability of an inverse problem for discontinuous Sturm–Liouville operators

In this paper, we consider the Sturm–Liouville equation with the jump conditions inside the interval (0,π). The inverse problem is studied, which consists in recovering operator coefficients from two spectra, corresponding to different boundary conditions. We prove the uniqueness theorem and provide necessary and sufficient conditions for solvability of the inverse problem. We also obtain the oscillation theorem for the eigenfunctions of the considered discontinuous boundary value problem.     

Essential spectrum of singular discrete linear Hamiltonian systems

The paper is concerned with the essential spectral points of singular discrete linear Hamiltonian systems. Several sufficient conditions for a real point to be in the essential spectrum are obtained in terms of the number of linearly independent square‐summable solutions of the corresponding homogeneous linear system, and a sufficient and necessary condition for a real point to be in the essential spectrum is given in terms of the number of linearly independent square‐summable solutions of the corresponding nonhomogeneous linear system. As a direct consequence, the corresponding results for singular higher‐order symmetric vector difference expressions are given.     

Non‐self‐adjoint Bessel and Sturm–Liouville boundary value problems in limit‐circle case

It is shown in the limit‐circle case that system of root functions of the non‐self‐adjoint maximal dissipative (accumulative) Bessel operator and its perturbation Sturm–Liouville operator form a complete system in the Hilbert space. Furthermore, asymptotic behavior of the eigenvalues of the maximal dissipative (accumulative) Bessel operators is investigated, and it is proved that system of root functions form a basis (Riesz and Bari bases) in the same Hilbert space. Copyright © 2014 John Wiley & Sons, Ltd.     

Applications of variational methods to Sturm–Liouville boundary‐value problem for fourth‐order impulsive differential equations

In this paper, we study Sturm–Liouville boundary‐value problem for fourth‐order impulsive differential equations. Applying variational methods, several new existence results are obtained. Copyright © 2013 John Wiley & Sons, Ltd.     

Stability of essential spectra of singular Sturm‐Liouville differential operators under perturbations small at infinity

This paper is concerned with the stability of essential spectra of singular Sturm‐Liouville differential operators with complex‐valued coefficients. It is proved that the essential spectrum of the corresponding minimal operator is preserved by perturbations small at infinity with respect to the unperturbed operator. Based on it, 1‐dimensional Schrödinger operators under local dilative perturbations are studied.     

Titchmarsh–Weyl m ‐functions for second‐order Sturm–Liouville problems with two singular endpoints

In this paper we consider some cases of Sturm–Liouville problems with two singular endpoints at x = 0 and x = ∞ which have a simple spectrum, and show that the simplicity of the spectrum can be built into the definition of a Titchmarsh–Weyl m ‐function from which the eigenfunction expansion can be constructed. The use of initial conditions at a point interior to the interval (0,∞) is avoided in favor of Frobenius solutions near the regular singular point x = 0. In contrast to the classical theory associated with a regular left endpoint, the growth behaviour of the associated spectral functions can be on the order of λ^β for any β ∈ (0,∞). Application of the theory to the Bessel equation on (0,∞) and to the radial part of the separated hydrogen atom on (0,∞) is given. In the case of the hydrogen atom a single Titchmarsh–Weyl m ‐function is obtained which completely describes both the discrete negative spectrum and the continuous positive spectrum. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sturm–Liouville operators in the half axis with shifted potentials

We consider Sturm–Liouville operators in the half axis generated by shifts of the potential and prove that Lebesgue measure is equivalent to a measure defined as an average of spectral measures which correspond to these operators. This is then used to obtain results on stability of spectral types under change of parameters such as boundary conditions, local perturbations, and shifts. In particular if for a set of shifts of positive measure the corresponding operators have α-singular, singular continuous and (or) point spectrum in a fixed interval, then this set of shifts has to be unbounded. Moreover, there are large sets of boundary conditions and local perturbations for which the corresponding operators enjoy the same property.     

Computing the indices of Sturm–Liouville eigenvalues for coupled boundary conditions (the EIGENIND-SLP codes)

The main purpose of the EIGENIND-SLP codes is to compute the indices of known eigenvalues of self-adjoint Sturm–Liouville problems with coupled boundary conditions (BCs). The spectrum of the problems can be unbounded from both below and above. Using some recent theoretical results, the computation is converted to that of the indices of the same eigenvalues for appropriate separated BCs, and is then carried out in terms of the Prüfer angle. The algorithm so generated and its implementation are discussed, and numerous examples are presented to illustrate the theoretical results and various aspects of the implementation.