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Spectral Analysis Of Differential Operators : Interplay Between Spectral And Oscillatory Properties.

Author: Fedor S Rofe-Beketov; Aleksandr M Kholkin
Publisher: Singapore : World Scientific, 2005.
Series: World Scientific monograph series in mathematics, v. 7.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
This is the first monograph devoted to the Sturm oscillatory theory for infinite systems of differential equations and its relations with the spectral theory. It aims to study a theory of self-adjoint problems for such systems, based on an elegant method of binary relations. Another topic investigated in the book is the behavior of discrete eigenvalues which appear in spectral gaps of the Hill operator and almost  Read more...
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Genre/Form: Electronic books
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Fedor S Rofe-Beketov; Aleksandr M Kholkin
ISBN: 9789812703453 9812703454 1281905828 9781281905826 9812562761 9789812562760
OCLC Number: 476064524
Description: 1 online resource (463 pages)
Contents: Cover --
Contents --
Foreword --
Preface --
Acknowledgments --
Introduction --
1. Relation Between Spectral and Oscillatory Properties for the Matrix Sturm-Liouville Problem --
1.1 Problem on a Finite Interval --
1.2 Problem on the Half-Line --
1.3 Bibliographical Comments --
1.3.1 An Example of the Operator of (1.1)(1.3) Type with the Purely Absolutely Continuous Spectrum --
1.3.2 Oscillatory Theory of Partial Differential Equations --
1.3.3 Bochner Integral --
2. Fundamental System of Solutions for an Operator Differential Equation with a Singular Boundary Condition --
2.1 Separated Self-Adjoint Boundary Conditions --
2.2 A Construction for the Fundamental Solution of the Boundary-Value Problem --
2.3 Self-consistency of the Fundamental Solution of the Self- Adjoint Problem and the Evolution of the Corresponding Hermitian Relation (or Lagrangian Plane in H @ H) --
2.4 A Different Construction for the Fundamental Solution of the Boundary-Value Problem --
2.5 Bibliographical Comments --
3. Dependence of the Spectrum of Operator Boundary Problems on Variations of a Finite or Semi-Infinite Interval --
3.1 Dependence of Eigenvalues and the Greatest Lower Bound of the Spectrum for a Semi-Bounded Below Differential Operator on Variations of the Interval --
3.2 Continuity and Monotonicity of the Greatest Lower Bound of the Essential Spectrum for Semi-Bounded Below Differential Operators --
3.3 Bibliographical Comments --
4. Relation Between Spectral and Oscillatory Properties for Operator Differential Equations of Arbitrary Order --
4.1 Oscillatory Theorem for Operator Differential Equations of Even Order --
4.2 Comparison and Alternation Theorems --
4.3 Multiplicative Representation of Positive Differential Operators and Its Applications --
4.4 Discrete Levels in the Gaps of the Essential Spectrum --
4.5 Bibliographical Comments --
4.5.1 Symplectic Interpretation of Sturm-Type Theorems and Their Operator-Theoretical Proofs --
5. Self-Adjoint Extensions of Systems of Differential Equations of Arbitrary Order on an Infinite Interval in the Absolutely Indefinite Case --
5.1 A Description of Self-Adjoint Extensions of Differential Operators of Arbitrary Order with Operator-Valued Coefficients on an Infinite Interval --
5.2 Parametrization of the Characteristic Operator Function --
5.3 Bibliographical Comments --
5.3.1 Everitt-Zettl Problem, Brusentsevs Example and Two Open Questions --
5.3.2 On the Deficiency Indices of Scalar Symmetric Dif- ferential Operators of General Kind on the Half-Axis (Solved and Unsolved Questions) --
5.3.3 Kogan-Rofe-Beketovs Asymptotic Theorems and Deficiency Indices of Symmetric Differential Operator --
5.3.4 R.C. Gilberts Class of Formally Self-Adjoint Ordi- nary Differential Operators Whose Deficiency Numbers Differ by an Arbitrary Pre-Assigned Positive Integer --
5.3.5 Deficiency Indices of Symmetric Differential Systems of the First and Arbitrary Orders and Some Open Questions --
5.3.6 Some Comments on Hilberts 21-st Problem and Bolibrukh Counterexample --
5.3.7 Some Comments on Section 5.2 --
5.3.8 Characteristic Properties of Weyl Solutions for the Sturm-Liouville and Dirac Equations. V.A. Marchenkos Theorems --
6. Discrete Levels in Spectral Gaps of Perturbed Schrodinger and Hill Operators --
6.1 Factorized Phase Matrix of the Perturbed System and the Discrete Spectrum in Gaps of the Continuous Spectrum --
6.2 Generalization of D Alembert-Liouville-Ostrogradsky For- mula and Its Application to Growth Estimates for Solutions of Canonical Almost Periodic Systems --
T$19058.
Series Title: World Scientific monograph series in mathematics, v. 7.

Abstract:

This is the first monograph devoted to the Sturm oscillatory theory for infinite systems of differential equations and its relations with the spectral theory. It aims to study a theory of self-adjoint problems for such systems, based on an elegant method of binary relations. Another topic investigated in the book is the behavior of discrete eigenvalues which appear in spectral gaps of the Hill operator and almost periodic Schrdinger operators due to local perturbations of the potential (e.g., modeling impurities in crystals). The book is based on results that have not been presented in other.

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