Discrete Sturm–Liouville problems, Jacobi matrices and Lagrange interpolation series

The close relationship between discrete Sturm–Liouville problems belonging to the so-called limit-circle case, the indeterminate Hamburger moment problem and the search of self-adjoint extensions of the associated semi-infinite Jacobi matrix is well known. In this paper, all these important topics are also related with associated sampling expansions involving analytic Lagrange-type interpolation series.     

Formal and precise derivation of the Green functions for a simple potential

In formal scattering theory, the Green functions are obtained as solutions of a distributional equation. In this paper, we use the Sturm–Liouville theory to compute the Green functions within a rigorous mathematical theory. We shall show that both the Sturm–Liouville theory and the formal treatment yield the same Green functions. We shall also show how the analyticity of the Green functions as functions of the energy keeps track of the so-called “incoming” and “outgoing” boundary conditions.     

M-matrix asymptotics for Sturm–Liouville problems on graphs

We consider a system formulation for Sturm–Liouville operators with formally self-adjoint boundary conditions on a graph. An M-matrix associated with the boundary value problem is defined and related to the matrix Prüfer angle associated with the system boundary value problem, and consequently with the boundary value problem on the graph. Asymptotics for the M-matrix are obtained as the eigenparameter tends to negative infinity. We show that the boundary conditions may be recovered, up to a unitary equivalence, from the M-matrix and that the M-matrix is a Herglotz function. This is the first in a series of papers devoted to the reconstruction of the Sturm–Liouville problem on a graph from its M-matrix.     

A twist condition and multiple solutions of unbounded self-adjoint operator equation with symmetries

By using the index theory for unbounded self-adjoint operator equations and the symmetric mountain pass theorem, we investigate the existence of multiple solutions for nonlinear operator equations with twist conditions. We prove an abstract theorem, and give some applications to first order Hamiltonian systems with Sturm–Liouville boundary conditions and delay differential equations.     

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).     

Oscillation properties for the equation of vibrating beam with irregular boundary conditions

In this paper we study the oscillatory properties for the eigenfunctions of some fourth-order eigenvalue problems, where the boundary conditions are irregular in the sense of the classification of [S. Janczewski, Oscillation theorems for the differential boundary value problems of the fourth order, Ann. of Math. 29 (1928) 521–542]. In this case, we show that these oscillatory properties are different from those of the Sturm–Liouville problem.     

Sturm–Liouville Problems with Coupled Boundary Conditions and Lagrange Interpolation Series

This paper is concerned with the application of the Kramer sampling theorem to Sturm–Liouville problems with coupled boundary conditions. The analysis is restricted to the case when the spectrum of the boundary value problem is simple. In all such cases, it is shown that Kramer analytic kernels can be defined and that each kernel has an associated analytic interpolation function to give the Lagrange interpolation series.     

On impulsive Sturm–Liouville operators with Coulomb potential and spectral parameter linearly contained in boundary conditions

The Sturm–Liouville problem with linear discontinuities is investigated in the case where an eigenparameter appears not only in a differential equation but also in boundary conditions. Properties and the asymptotic behavior of spectral characteristics are studied for the Sturm–Liouville operators with Coulomb potential that have discontinuity conditions inside a finite interval. Moreover, the Weyl function for this problem is defined and uniqueness theorems are proved for a solution of the inverse problem with respect to this function.     

On Sturm–Liouville boundary value problems for second-order nonlinear functional finite difference equations

Motivated by the interesting paper [I. Karaca, Discrete third-order three-point boundary value problem, J. Comput. Appl. Math. 205 (2007) 458–468], this paper is concerned with a class of boundary value problems for second-order functional difference equations. Sufficient conditions for the existence of at least one solution of a Sturm–Liouville boundary value problem for second-order nonlinear functional difference equations are established. We allow f to be at most linear, superlinear or sublinear in obtained results.     

Method of Immersion and Its Parametric Realization for the Korteweg–De Vries Equation

By using the method of immersion (imbedding) proposed in the author's previous works, we describe the space S of initial conditions of the Cauchy problem for the general differential Korteweg–de Vries equation. The space S is called a stationary soliton Korteweg–de Vries manifold because "stationary projections" of solitons fall into the space S. In addition, we introduce the notion of a space of Sturm–Liouville operators over a soliton Korteweg–de Vries manifold. For real functions and parameters, we formulate the spectral theorem for a commutative Lax pair over a real stationary soliton Korteweg–de Vries manifold.     

Sturm–Liouville Problem for Second Order Ordinary Differential Equations Across Resonance

This paper is concerned with the existence and uniqueness of solutions to the Sturm–Liouville boundary value problem across resonance. By using optimal control theory, we present some global optimality results about the unique solvability for the Sturm–Liouville problem.     

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.     

Dissipative Sturm–Liouville Operators in Limit-Point Case

Dissipative singular Sturm–Liouville operators are studied in the Hilbert space L_w²[a,b) (–<a<b), that the extensions of a minimal symmetric operator in Weyls limit-point case. We construct a selfadjoint dilation of the dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and define its characteristic function in terms of the Titchmarsh–Weyl function of a selfadjoint operator. Finally, in the case when the Titchmarsh–Weyl function of the selfadjoint operator is a meromorphic in complex plane, we prove theorems on completeness of the system of eigenfunctions and associated functions of the dissipative Sturm–Liouville operators. Mathematics Subject Classifications (2000) 47A20, 47A40, 47A45, 34B20, 34B44, 34L10.     

Inverse Problems for Sturm–Liouville Operators on Noncompact Trees

We study Sturm–Liouville differential operators on noncompact graphs without cycles (i.e., on trees) with standard matching conditions in internal vertices. First we establish properties of the spectral characteristics and then we investigate the inverse problem of recovering the operator from the so-called Weyl vector. For this inverse problem we prove a uniqueness theorem and propose a procedure for constructing the solution using the method of spectral mappings. Received: February 13, 2007.     

Several existence theorems of nonlinear m-point boundary value problem for p-Laplacian dynamic equations on time scales

In this paper, several existence theorems of positive solutions are established for nonlinear m-point boundary value problem for p-Laplacian dynamic equations on time scales, as an application, an example to demonstrate our results is given. The conditions we used in the paper are different from those in [H.R. Sun, W.T. Li, Positive solutions for nonlinear three-point boundary value problems on time scales, J. Math. Anal. Appl. 299 (2004) 508–524; H.R. Sun, W.T. Li, Positive solutions for nonlinear m-point boundary value problems on time scales, Acta Math. Sinica 49 (2006) 369–380 (in Chinese); Y. Wang, C. Hou, Existence of multiple positive solutions for one-dimensional p-Laplacian, J. Math. Anal. Appl. 315 (2006) 144–153; Y. Wang, W. Ge, Positive solutions for multipoint boundary value problems with one-dimensional p-Laplacian, Nonlinear Appl. 66 (6) (2007) 1246–1256].     

Canonical forms of self-adjoint boundary conditions for differential operators of order four

Canonical forms of regular self-adjoint boundary conditions for differential operators are well known in the second order i.e. Sturm–Liouville case. In this paper we find canonical forms for fourth order self-adjoint boundary conditions.     

Multiplicity results for Sturm–Liouville boundary value problems

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

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.     

Two-Interval Sturm–Liouville Operators in Direct Sum Spaces with Inner Product Multiples

In direct sum spaces with inner product multiples, we study two-interval Sturm–Liouville problems. For singular problems, we generate self-adjoint realizations for boundary conditions with any real coupling matrix whose determinant is positive. This contrasts with the usual theory which requires the coupling matrix to have determinant one. This paper is in final form and no version of it will be submitted for publication elsewhere. Received: September 28, 2006.