A symmetric generalization of Sturm–Liouville problems in discrete spaces

Classical Sturm–Liouville problems of a discrete variable are extended for symmetric functions such that the corresponding solutions preserve the orthogonality property. Some generic illustrative examples are given in this sense.     

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.     

A symmetric generalization of Sturm–Liouville problems in q-difference spaces

Classical Sturm–Liouville problems of q-difference variables are extended for symmetric discrete functions such that the corresponding solutions preserve the orthogonality property. Some illustrative examples are given in this sense.     

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.     

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.     

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.     

Non-localization of eigenfunctions for Sturm–Liouville operators and applications

In this article, we investigate a non-localization property of the eigenfunctions of Sturm–Liouville operators $A_a=?{?}_{xx}+a(?)\mathrm{Id}$ with Dirichlet boundary conditions, where $a(?)$ runs over the bounded nonnegative potential functions on the interval $(0,L)$ with $L>0$. More precisely, we address the extremal spectral problem of minimizing the $L^2$-norm of a function $e(?)$ on a measurable subset ω of $(0,L)$, where $e(?)$ runs over all eigenfunctions of $A_a$, at the same time with respect to all subsets ω having a prescribed measure and all $L^{\infty }$ potential functions $a(?)$ having a prescribed essentially upper bound. We provide some existence and qualitative properties of the minimizers, as well as precise lower and upper estimates on the optimal value. Several consequences in control and stabilization theory are then highlighted.     

On degenerate boundary conditions in the Sturm–Liouville problem

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

On Fourier Series in Generalized Eigenfunctions of a Discrete Sturm–Liouville Operator

For semicontinuous summation methods generated by Λ = {λ_n(h)} (n = 0, 1, 2,...; h > 0) of Fourier series in eigenfunctions of a discrete Sturm–Liouville operator of class B, some results on the uniform a.e. behavior of Λ-means are obtained. The results are based on strong- and weak-type estimates of maximal functions. As a consequence, some statements on the behavior of the summation methods generated by the exponential means λ_n(h) = exp(?u^α(n)h) are obtained. An application to a generalized heat equation is given.     

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.     

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.