Using Time Scales to Study Multi-Interval Sturm–Liouville Problems with Interface Conditions

We consider a Sturm–Liouville problem defined on multiple intervals with interface conditions. The existence of a sequence of eigenvalues is established and the zero counts of associated eigenfunctions are determined. Moreover, we reveal the continuous and discontinuous nature of the eigenvalues on the boundary condition. The approach in this paper is different from those in the literature: We transfer the Sturm–Liouville problem with interface conditions to a Sturm–Liouville problem on a time scale without interface conditions and then apply the Sturm–Liouville theory for equations on time scales. In this way, we are able to investigate the problem in a global view. Consequently, our results cover the cases when the potential function in the equation is not strictly greater than zero and when the domain consists of an infinite number of intervals.     

On the Uniqueness of the Solution of the Inverse Sturm–Liouville Problem with Nonseparated Boundary Conditions on a Geometric Graph

For the first time, the inverse Sturm–Liouville problem with nonseparated boundary conditions is studied on a star-shaped geometric graph with three edges. It is shown that the Sturm–Liouville problem with general boundary conditions cannot be uniquely reconstructed from four spectra. Nonseparated boundary conditions are found for which a uniqueness theorem for the solution of the inverse Sturm–Liouville problem is proved. The spectrum of the boundary value problem itself and the spectra of three auxiliary problems are used as reconstruction data. It is also shown that the Sturm–Liouville problem with these nonseparated boundary conditions can be uniquely recovered if three spectra of auxiliary problems are used as reconstruction data and only five of its eigenvalues are used instead of the entire spectrum of the problem.     

Note on a nonlocal Sturm–Liouville problem with both right and left fractional derivatives

In this paper, we research the geometric multiplicity of eigenvalues for a nonlocal Sturm–Liouville eigenvalue problem. To this end, we study the uniqueness of solutions for a nonlocal Sturm–Liouville problem under some initial value conditions.     

Resolvent Operator and Self-Adjointness of Sturm–Liouville Operators with a Finite Number of Transmission Conditions

In this paper, we consider a Sturm–Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at a finite number of interior points. We introduce a Hilbert space formulation such that the problem under consideration can be interpreted as an eigenvalue problem for a suitable self-adjoint linear operator. We construct Green function of the problem and resolvent operator. We establish the self-adjointness of the discontinuous Sturm–Liouville operator.     

Ambarzumyan‐type theorem with polynomially dependent eigenparameter

In this paper, we are concerned with the inverse Sturm–Liouville problem with polynomially dependent eigenparameter in discontinuity and boundary conditions. By using a self‐adjoint operator‐theoretic interpretation for this sort of problem, Ambarzumyan theorem is provided for the mentioned Sturm–Liouville operator. Copyright © 2014 John Wiley & Sons, Ltd.     

Matrix representations of Sturm‐Liouville problems with coupled eigenparameter‐dependent boundary conditions and transmission conditions

We study matrix representations of Sturm‐Liouville problems with coupled eigenparameter‐dependent boundary conditions and transmission conditions. Meanwhile, given any matrix eigenvalue problem with coupled eigenparameter‐dependent boundary conditions and transmission conditions, we construct a class of Sturm‐Liouville problems with given boundary conditions and transmission conditions such that they have the same eigenvalues.     

On a Sturm–Liouville type problem with retarded argument

In this work, a Sturm–Liouville‐type problem with retarded argument, which contains spectral parameter in the boundary conditions and with transmission conditions at the point of discontinuity are investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions. Copyright © 2012 John Wiley & Sons, Ltd.     

A hierarchy of Sturm–Liouville problems

Sturm–Liouville equations will be considered where the boundary conditions depend rationally on the eigenvalue parameter. Such problems apply to a variety of engineering situations, for example to the stability of rotating axles. Classesof these problems will be isolated with a rather rich spectral structure, for example oscillation, comparison and completeness properties analogous to thoseof the ‘usual’ Sturm–Liouville problem which has constant boundary conditions.In fact it will be shown how these classes can be converted into each other, andinto the ‘usual’ Sturm–Liouville problem, by means of transformations preserving all but finitely many eigenvalues. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

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.     

Spectral theory of Sturm–Liouville difference operators

We present several classes of explicit self-adjoint Sturm–Liouville difference operators with either a non-Hermitian leading coefficient function, or a non-Hermitian potential function, or a non-definite weight function, or a non-self-adjoint boundary condition. These examples are obtained using a general procedure for constructing difference operators realizing discrete Sturm–Liouville problems, and the minimum conditions for such difference operators to be self-adjoint with respect to a natural quadratic form. It is shown that a discrete Sturm–Liouville problem admits a difference operator realization if and only if it does not have all complex numbers as eigenvalues. Spectral properties of self-adjoint Sturm–Liouville difference operators are studied. In particular, several eigenvalue comparison results are proved.     

A class of second‐order differential operators with eigenparameter‐dependent boundary and transmission conditions

In this study, we investigate a Sturm–Liouville type problem with eigenparameter‐dependent boundary conditions and eigenparameter‐dependent transmission conditions. By establishing a new self‐adjoint operator A associated with the problem, we construct fundamental solutions and obtain asymptotic formulae for its eigenvalues and fundamental solutions. Also we investigate some properties of its spectrum. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Borg-type theorem for the missing eigenvalue problem

In this paper, we discuss the inverse spectral problem for Sturm–Liouville operators for the missing eigenvalue problem. We show that a Borg-type theorem for the missing eigenvalue problem of the Sturm–Liouville operator holds by the Weyl $m$-function.     

Solvability and stability of the inverse Sturm–Liouville problem with analytical functions in the boundary condition

The paper deals with the Sturm–Liouville eigenvalue problem with the Dirichlet boundary condition at one end of the interval and with the boundary condition containing entire functions of the spectral parameter at the other end. We study the inverse problem, which consists in recovering the potential from a part of the spectrum. This inverse problem generalizes partial inverse problems on finite intervals and on graphs and also the inverse transmission eigenvalue problem. We obtain sufficient conditions for global solvability of the studied inverse problem, which prove its local solvability and stability. In addition, application of our main results to the partial inverse Sturm–Liouville problem on the star-shaped graph is provided.     

On degenerate boundary conditions in the Sturm–Liouville problem

We describe all degenerate boundary conditions in the homogeneous Sturm–Liouville problem.     

Existence Conditions of Negative Eigenvalues in the Regular Sturm–Liouville Boundary Value Problem and Explicit Expressions for Their Number

Computational Mathematics and Mathematical Physics - For the regular Sturm–Liouville boundary value problem with general nonseparated self-adjoint boundary conditions, conditions for the...     

On the eigenvalues of a nonlinear spectral problem

We consider a nonlinear eigenvalue problem of the Sturm–Liouville type with conditions of the third kind, which describes the propagation of polarized electromagnetic waves in a plane dielectric waveguide. The equation is nonlinear in the unknown function, and the boundary conditions depend on the spectral parameter nonlinearly. We obtain an equation for the spectral parameter and formulas for the zeros of the eigenfunctions and show that the problem has at most finitely many isolated eigenvalues.     

Sampling theory for Sturm–Liouville problem with boundary and transmission conditions containing an eigenparameter

In this paper, we derive the sampling theorem associated with a Sturm–Liouville problem which has two points of discontinuity and contains an eigenparameter in a boundary condition and also two transmission conditions. We establish briefly spectral properties of the problem, and then, we prove the sampling theorem associated with the problem.     

Inverse Sturm–Liouville problem on equilateral regular tree

In this article, we consider a spectral problem generated by the Sturm–Liouville equation on the edges of an equilateral regular tree. It is assumed that the Dirichlet boundary conditions are imposed at the pendant vertices and continuity and Kirchhoff's conditions at the interior vertices. The potential in the Sturm–Liouville equations, the same on each edge, is real, symmetric with respect to the middle of an edge and belongs to L ₂(0,?a) where a is the length of an edge. Conditions are obtained on a sequence of real numbers necessary and sufficient to be the spectrum of the considered spectral problem.