Format:
Online-Ressource (VIII, 216 p. 1 illus. in color)
,
digital
Edition:
Online-Ausg. Springer eBook Collection. Mathematics and Statistics Electronic reproduction; Available via World Wide Web
ISBN:
9783034805285
Series Statement:
Operator Theory: Advances and Applications 230
Content:
Periodic differential operators have a rich mathematical theory as well as important physical applications. They have been the subject of intensive development for over a century and remain a fertile research area. This book lays out the theoretical foundations and then moves on to give a coherent account of more recent results, relating in particular to the eigenvalue and spectral theory of the Hill and Dirac equations. The book will be valuable to advanced students and academics both for general reference and as an introduction to active research topics.
Content:
Periodic differential operators have a rich mathematical theory as well as important physical applications. They have been the subject of intensive development for over a century and remain a fertile research area. This book lays out the theoretical foundations and then moves on to give a coherent account of more recent results, relating in particular to the eigenvalue and spectral theory of the Hill and Dirac equations. The book will be valuable to advanced students and academics both for general reference and as an introduction to active research topics
Note:
Description based upon print version of record
,
Periodic Differential Operators; Contents; Preface; Chapter 1 Floquet Theory; 1.1 Introduction; 1.2 Preliminaries on ordinary differential systems; 1.3 Periodic first-order systems; 1.4 The discriminant and stability; 1.5 Hill's equation and periodic Dirac systems; 1.6 Functional properties of Hill's discriminant; 1.7 The Mathieu equation; 1.8 Periodic, semi-periodic and twisted boundary-value problems; 1.9 Appendix: Rofe-Beketov's formula; 1.10 Chapter notes; Chapter 2 Oscillations; 2.1 Introduction; 2.2 The Prüfer transform; 2.3 The boundary-value problem with separated boundary conditions
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2.4 The rotation number2.5 Zeros of solutions of Hill's equation; 2.6 The upper end-points of the stability intervals; 2.7 A step-function example; 2.8 Even coefficients; 2.9 Comparison of eigenvalues; 2.10 Least eigenvalues; 2.11 Chapter notes; Chapter 3 Asymptotics; 3.1 Introduction; 3.2 Prüfer transformation formulae; 3.3 The coefficient w; 3.4 Titchmarsh's asymptotic formula; 3.5 Differentiable q; 3.6 Length of the instability intervals; 3.7 The Mathieu equation; 3.8 Asymptotic formulae for solutions; 3.9 Absence of instability intervals
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3.10 Absence of all but N finite instability intervals3.11 Absence of odd instability intervals; 3.12 All instability intervals non-vanishing; 3.13 Chapter notes; Chapter 4 Spectra; 4.1 Introduction; 4.2 Regular boundary-value problems; 4.3 The spectral function for the half-line problem; 4.4 Self-adjoint half-line operators; 4.5 The spectrum of the periodic boundary-value problem on the half-line; 4.6 The spectral matrix for the full-line problem; 4.7 The spectrum of the full-line periodic problem; 4.8 Oscillations and spectra; 4.9 Bounded solutions and the absolutely continuous spectrum
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4.10 Chapter notesChapter 5 Perturbations; 5.1 Introduction; 5.2 Spectral bands; 5.3 Gap eigenvalues; 5.4 Critical coupling constants; 5.5 Eigenvalue asymptotics; 5.6 Chapter notes; Bibliography; Index
,
Electronic reproduction; Available via World Wide Web
Additional Edition:
ISBN 9783034805278
Language:
English
Subjects:
Mathematics
DOI:
10.1007/978-3-0348-0528-5
URL:
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