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2005 | Buch

Sturm’s 1836 Oscillation Results Evolution of the Theory Kapitel lesen Erstes Kapitel lesen

Sturm-Liouville Theory

Past and Present

herausgegeben von: Werner O. Amrein, Andreas M. Hinz, David P. Pearson

Verlag: Birkhäuser Basel

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Inhaltsverzeichnis

Frontmatter
Sturm’s 1836 Oscillation Results Evolution of the Theory
Abstract
We examine how Sturm’s oscillation theorems on comparison, separation, and indexing the number of zeros of eigenfunctions have evolved. It was Bôcher who first put the proofs on a rigorous basis, and major tools of analysis where introduced by Picone, Prüfer, Morse, Reid, and others. Some basic oscillation and disconjugacy results are given for the second-order case. We show how the definitions of oscillation and disconjugacy have more than one interpretation for higher-order equations and systems, but it is the definitions from the calculus of variations that provide the most fruitful concepts; they also have application to the spectral theory of differential equations. The comparison and separation theorems are given for systems, and it is shown how they apply to scalar equations to give a natural extension of Sturm’s second-order case. Finally we return to the second-order case to show how the indexing of zeros of eigenfunctions changes when there is a parameter in the boundary condition or if the weight function changes sign.
Don Hinton
Sturm Oscillation and Comparison Theorems
Abstract
This is a celebratory and pedagogical discussion of Sturm oscillation theory. Included is the discussion of the difference equation case via determinants and a renormalized oscillation theorem of Gesztesy, Teschl, and the author.
Barry Simon
Charles Sturm and the Development of Sturm-Liouville Theory in the Years 1900 to 1950
Abstract
The first joint publication by Sturm and Liouville in 1837 introduced the general theory of Sturm-Liouville differential equations.
This present paper is concerned with the remarkable development in the theory of Sturm-Liouville boundary value problems, which took place during the years from 1900 to 1950.
Whilst many mathematicians contributed to Sturm-Liouville theory in this period, this manuscript is concerned with the early work of Sturm and Liouville (1837) and then the contributions of Hermann Weyl (1910), A.C. Dixon (1912), M.H. Stone (1932) and E.C. Titchmarsh (1940 to 1950).
The results of Weyl and Titchmarsh are essentially derived within classical, real and complex mathematical analysis. The results of Stone apply to examples of self-adjoint operators in the abstract theory of Hilbert spaces and in the theory of ordinary linear differential equations.
In addition to giving some details of these varied contributions an attempt is made to show the interaction between these two different methods of studying Sturm-Liouville theory.
W. Norrie Everitt
Spectral Theory of Sturm-Liouville Operators Approximation by Regular Problems
Abstract
It is the aim of this article to present a brief overview of the theory of Sturm-Liouville operators, self-adjointness and spectral theory: minimal and maximal operators, Weyl’s alternative (limit point/limit circle case), deficiency indices, self-adjoint realizations, spectral representation.
The main part of the lecture will be devoted to the method of proving spectral results by approximating singular problems by regular problems: calculation/approximation of the discrete spectrum as well as the study of the absolutely continuous spectrum. For simplicity, most results will be presented only for the case where one end point is regular, but they can be extended to the general case, as well as to Dirac systems, to discrete operators, and (partially) to ordinary differential operators of arbitrary order.
Joachim Weidmann
Spectral Theory of Sturm-Liouville Operators on Infinite Intervals: A Review of Recent Developments
Abstract
This review discusses some of the central developments in the spectral theory of Sturm-Liouville operators on infinite intervals over the last thirty years or so. We discuss some of the natural questions that occur in this framework and some of the main models that have been studied.
Yoram Last
Asymptotic Methods in the Spectral Analysis of Sturm-Liouville Operators
Abstract
We consider the relationship between the asymptotic behavior of solutions of the singular Sturm-Liouville equation and spectral properties of the corresponding self-adjoint operators. In particular, we review the main features of the theory of subordinacy by considering two standard cases, the half-line operator on L2([0, ∞)) and the full-line operator on L2(ℝ). It is assumed that the coefficient function q is locally integrable, that 0 is a regular endpoint in the half-line case, and that Weyl’s limit point case holds at the infinite endpoints. We note some consequences of the theory for the well-known informal characterization of the spectrum in terms of bounded solutions. We also consider extensions of the theory to related differential and difference operators, and discuss its application, in conjunction with other asymptotic methods, to some typical problems in spectral analysis.
Daphne Gilbert
The Titchmarsh-Weyl Eigenfunction Expansion Theorem for Sturm-Liouville Differential Equations
Abstract
This paper involves a revisit to the original works of Hermann Weyl in 1910 and of Edward Charles Titchmarsh in 1941, concerning Sturm-Liouville theory and the corresponding eigenfunction expansions.
For this account the essential results of Weyl concern the regular, limit-circle and limit-point classifications of Sturm-Liouville differential equations; the eigenfunction expansion theory from Titchmarsh is based on classical function theory methods, in particular complex function theory.
The eigenfunction expansion theory presented in this paper is based on the Titchmarsh-Weyl m-coefficient; the proofs are essentially in classical function theory but are related to operator theoretic methods in Hilbert space.
One important innovation here is that for the Sturm-Liouville problem considered on the open interval (a,b), the endpoint a can be classified as either regular or limit-circle, whilst the endpoint b can be regular, limit-circle or limit-point; nevertheless it is shown that these conditions lead to the definition of a single Titchmarsh-Weyl m-coefficient. From this coefficient the complex function theory methods of Titchmarsh provide a guide to a new proof of the general eigenfunction expansion theorem.
Christer Bennewitz, W. Norrie Everitt
Sturm’s Theorems on Zero Sets in Nonlinear Parabolic Equations
Abstract
We present a survey on applications of Sturm’s theorems on zero sets for linear parabolic equations, established in 1836, to various problems including reaction-diffusion theory, curve shortening and mean curvature flows, symplectic geometry, etc. The first Sturm theorem, on nonincrease in time of the number of zeros of solutions to one-dimensional heat equations, is shown to play a crucial part in a variety of existence, uniqueness and asymptotic problems for a wide class of quasilinear and fully nonlinear equations of parabolic type. The survey covers a number of the results obtained in the last twenty-five years and establishes links with earlier ones and those in the ODE area.
Victor A. Galaktionov, Petra J. Harwin
A Survey of Nonlinear Sturm-Liouville Equations
Abstract
This note gives a brief survey of existence, uniqueness and bifurcation results for nonlinear Sturm-Liouville equations. Early in 1960, Nehari made an interesting proposal to study solutions with a prescribed number of nodes. His ideas have had a great influence on critical point theory as a branch of the calculus of variations. Rabinowitz established a global bifurcation theorem based on the nodal properties of solutions. Some results on bifurcation from the lowest point of the continuous spectrum will also be discussed.
Chao-Nien Chen
Boundary Conditions and Spectra of Sturm-Liouville Operators
Abstract
This is a discussion of some aspects of the relation between boundary conditions and spectra of Sturm-Liouville operators. It is intended to review results which show how the spectrum behaves when the boundary condition changes. The absolutely continuous part will normally be stable and the more interesting problems concern the behavior of the singular part and coexistence of different spectral types.
Rafael del Río
Uniqueness of the Matrix Sturm-Liouville Equation given a Part of the Monodromy Matrix, and Borg Type Results
Abstract
Uniqueness of the matrix Sturm-Liouville equation is investigated, given a part of its monodromy matrix. Generalizations of Borg’s theorem and the Hochstadt-Lieberman result for the matrix Sturm-Liouville equation are presented.
Mark M. Malamud
A Catalogue of Sturm-Liouville Differential Equations
Abstract
This catalogue commences with sections devoted to a brief summary of Sturm-Liouville theory including some details of differential expressions and equations, Hilbert function spaces, differential operators, classification of interval endpoints, boundary condition functions and the Liouville transform.
There follows a collection of more than 50 examples of Sturm-Liouville differential equations; many of these examples are connected with well-known special functions, and with problems in mathematical physics and applied mathematics.
For most of these examples the interval endpoints are classified within the relevant Hilbert function space, and boundary condition functions are given to determine the domains of the relevant differential operators. In many cases the spectra of these operators are given.
The author is indebted to many colleagues who have responded to requests for examples and who checked successive drafts of the catalogue.
W. Norrie Everitt
Backmatter
Metadaten
Titel
Sturm-Liouville Theory
herausgegeben von
Werner O. Amrein
Andreas M. Hinz
David P. Pearson
Copyright-Jahr
2005
Verlag
Birkhäuser Basel
Electronic ISBN
978-3-7643-7359-7
Print ISBN
978-3-7643-7066-4
DOI
https://doi.org/10.1007/3-7643-7359-8

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