Spectral Theory of Ordinary Differential Operators
Published on 6. May 1987
Book
Paperback/Softback
VIII, 304 pages
978-3-540-17902-3 (ISBN)
Description
These notes will be useful and of interest to mathematicians and physicists active in research as well as for students with some knowledge of the abstract theory of operators in Hilbert spaces. They give a complete spectral theory for ordinary differential expressions of arbitrary order n operating on -valued functions existence and construction of self-adjoint realizations via boundary conditions, determination and study of general properties of the resolvent, spectral representation and spectral resolution. Special attention is paid to the question of separated boundary conditions, spectral multiplicity and absolutely continuous spectrum. For the case nm=2 (Sturm-Liouville operators and Dirac systems) the classical theory of Weyl-Titchmarch is included. Oscillation theory for Sturm-Liouville operators and Dirac systems is developed and applied to the study of the essential and absolutely continuous spectrum. The results are illustrated by the explicit solution of a number of particular problems including the spectral theory one partical Schrödinger and Dirac operators with spherically symmetric potentials. The methods of proof are functionally analytic wherever possible.
More details
Series
Edition
1987 ed.
Language
English
Place of publication
Berlin
Germany
Publishing group
Springer Berlin
Target group
Professional and scholarly
Research
Illustrations
VIII, 304 p.
Dimensions
Height: 235 mm
Width: 155 mm
Thickness: 17 mm
Weight
476 gr
ISBN-13
978-3-540-17902-3 (9783540179023)
DOI
10.1007/BFb0077960
Schweitzer Classification
Content
Formally self-adjoint differential expressions.- Appendix to section 1: The separation of the Dirac operator.- Fundamental properties and general assumptions.- Appendix to section 2: Proof of the Lagrange identity for n>2.- The minimal operator and the maximal operator.- Deficiency indices and self-adjoint extensions of T0.- The solutions of the inhomogeneous differential equation (?-?)u=f; Weyl's alternative.- Limit point-limit circle criteria.- Appendix to section 6: Semi-boundedness of Sturm-Liouville type operators.- The resolvents of self-adjoint extensions of T0.- The spectral representation of self-adjoint extensions of T0.- Computation of the spectral matrix ?.- Special properties of the spectral representation, spectral multiplicities.- L2-solutions and essential spectrum.- Differential operators with periodic coefficients.- Appendix to section 12: Operators with periodic coefficients on the half-line.- Oscillation theory for regular Sturm-Liouville operators.- Oscillation theory for singular Sturm-Liouville operators.- Essential spectrum and absolutely continuous spectrum of Sturm-Liouville operators.- Oscillation theory for Dirac systems, essential spectrum and absolutely continuous spectrum.- Some explicitly solvable problems.