Anton Zettl

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Given any self-adjoint realization S of a singular Sturm-Liouville (S-L) problem, it is possible to construct a sequence {S r } of regular S-L problems with the properties (i) every point of the spectrum of S is the limit of a sequence of eigenvalues from the spectrum of the individual members of {S r } (ii) in the case when S is regular or limit-circle at(More)
The eigenvalues of linear, regular, two point boundary value problems depend continuously on the problem. In the important self-adjoint case studied by Naimark and Weidmann this dependence is differentiable and the derivatives of the eigenvalues with respect to a given parameter: an endpoint, a boundary condition, a coefficient, or the weight function, are(More)
We consider some geometric aspects of regular Sturm-Liouville problems. First, we clarify a natural geometric structure on the space of boundary conditions. This structure is the base for studying the dependence of Sturm-Liouville eigenvalues on the boundary condition, and reveals many new properties of these eigenvalues. In particular, the eigenvalues for(More)
Singular boundary conditions are formulated for non-selfadjoint Sturm-Liouville problems which are limit-circle in a very general sense. The characteristic determinant is constructed and it is shown that it can be used to extend the Birkhoff theory for so called 'Birkhoff regular boundary conditions' to the singular case. This is illustrated for a class of(More)
The generalized Sturm-Liouville problems in this paper stem from the ideas of Christ Shubin and Stolz, based on introducing singularities at a countable number of regular points on the real line. This idea is generalized to the introduction of a countable number of regular or limit-circle singular points. These results are shown to link with the work of(More)
SLEIGN is a software package for the computation of eigenvalues and eigenfunction:s of regular and singular Sturm-Liouville boundary value problems, The package is a modification and extension of a code with the same name developed by Bailey, Gordon, and Shampinej which is described in ACM Z'OMS 4 (1978), 193-208. The modifications and extensions include(More)
The eigenvalues of Sturm-Liouville (SL) problems depend not only continuously but smoothly on boundary points. The derivative of the nth eigenvalue as a function of an endpoint satisfies a first order differential equation. This for arbitrary (separated or coupled) self-adjoint regular boundary conditions. In addition, as the length of the interval shrinks(More)
The SLEIGN2 code is based on the ideas and methods of the original SLEIGN code of 1979. The main purpose of the SLEIGN2 code is to compute eigenvalues and eigenfunctions of regular and singular self-adjoint Sturm-Liouville problems, with both separated and coupled boundary conditions, and to approximate the continuous spectrum in the singular case. The code(More)