Some observations and remarks on differential operators generated by first-order boundary value problems

AbstractThis paper deals with the study of the set of all self-adjoint differential operators which are generated from first-order, linear, ordinary boundary value problems. These operators are defined on a weighted Hilbert function space and are examined as an application of the result obtained by Everitt and Markus in their paper in 1997. An investigation is given so that first-order self-adjoint boundary value problems are transformed to a study of the nature of the spectrum of associated self-adjoint operators. However, the analysis of this paper is restricted to consideration of conditions under which the spectral properties of these operators yield a discrete spectrum, and consequently to the determination of conditions under which the construction of Kramer analytic kernels, from the above boundary value problems, can be accomplished

The Zero-Removing Property and Lagrange-Type Interpolation Series

The classical Kramer sampling theorem, which provides a method for obtaining orthogonal sampling formulas, can be formulated in a more general nonorthogonal setting. In this setting, a challenging problem is to characterize the situations when the obtained nonorthogonal sampling formulas can be expressed as Lagrange-type interpolation series. In this article a necessary and sufficient condition is given in terms of the zero removing property. Roughly speaking, this property concerns the stability of the sampled functions on removing a finite number of their zeros

Askey-Wilson Type Functions, With Bound States

The two linearly independent solutions of the three-term recurrence relation of the associated Askey-Wilson polynomials, found by Ismail and Rahman in [22], are slightly modified so as to make it transparent that these functions satisfy a beautiful symmetry property. It essentially means that the geometric and the spectral parameters are interchangeable in these functions. We call the resulting functions the Askey-Wilson functions. Then, we show that by adding bound states (with arbitrary weights) at specific points outside of the continuous spectrum of some instances of the Askey-Wilson difference operator, we can generate functions that satisfy a doubly infinite three-term recursion relation and are also eigenfunctions of $q$-difference operators of arbitrary orders. Our result provides a discrete analogue of the solutions of the purely differential version of the bispectral problem that were discovered in the pioneering work [8] of Duistermaat and Gr\"unbaum.Comment: 42 pages, Section 3 moved to the end, minor correction

A conjecture on Exceptional Orthogonal Polynomials

Exceptional orthogonal polynomial systems (X-OPS) arise as eigenfunctions of Sturm-Liouville problems and generalize in this sense the classical families of Hermite, Laguerre and Jacobi. They also generalize the family of CPRS orthogonal polynomials. We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical system by a Darboux-Crum transformation. We give a proof of this conjecture for codimension 2 exceptional orthogonal polynomials (X2-OPs). As a by-product of this analysis, we prove a Bochner-type theorem classifying all possible X2-OPS. The classification includes all cases known to date plus some new examples of X2-Laguerre and X2-Jacobi polynomials

Inverse problems for Sturm-Liouville equations with boundary conditions linearly dependent on the spectral parameter from partial information

[[abstract]]Abstract.In this paper, we study the inverse spectral problems for Sturm–Liouville equations with boundary conditions linearly dependent on the spectral parameter and show that the potential of such problem can be uniquely determined from partial information on the potential and parts of two spectra, or alternatively, from partial information on the potential and a subset of pairs of eigenvalues and the normalization constants of the corresponding eigenvalues.[[notice]]補正完畢[[journaltype]]國外[[incitationindex]]SCI[[ispeerreviewed]]Y[[booktype]]紙本[[booktype]]電子版[[countrycodes]]DE

Spectral Asymptotics for Perturbed Spherical Schr\"odinger Operators and Applications to Quantum Scattering

We find the high energy asymptotics for the singular Weyl--Titchmarsh m-functions and the associated spectral measures of perturbed spherical Schr\"odinger operators (also known as Bessel operators). We apply this result to establish an improved local Borg-Marchenko theorem for Bessel operators as well as uniqueness theorems for the radial quantum scattering problem with nontrivial angular momentum.Comment: 20 page

Interpolation theory and first-order boundary value problems

This paper discusses the connection between Kramer analytic kernels derived from first-order, linear, ordinary boundary value problems represented by self-adjoint differential operators and one form of the Lagrange interpolation formula, and treats the dual formulation of the sampling process, that of interpolation. In following the kernel construction results obtained by the authors in a previous paper in 2002, the results in this successor paper complete the aimed project by showing that each of these Kramer analytic kernels has an associated analytic interpolation function to give the Lagrange interpolation series. © 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim