Spectral Theory of Ordinary Differential Operators

Joachim Weidmann (auth.)354017902X, 9783540179023

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.

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Table of contents :
Introduction….Pages 1-6
Formally self-adjoint differential expressions….Pages 7-15
Appendix to section 1: The separation of the Dirac operator….Pages 16-22
Fundamental properties and general assumptions….Pages 23-35
Appendix to section 2: Proof of the Lagrange identity for n>2….Pages 35-40
The minimal operator and the maximal operator….Pages 41-51
Deficiency indices and self-adjoint extensions of T 0 ….Pages 52-71
The solutions of the inhomogeneous differential equation (τ-λ)u=f; Weyl’s alternative….Pages 72-87
Limit point-limit circle criteria….Pages 88-103
Appendix to section 6: Semi-boundedness of Sturm-Liouville type operators….Pages 104-109
The resolvents of self-adjoint extensions of T 0 ….Pages 110-125
The spectral representation of self-adjoint extensions of T 0 ….Pages 126-139
Computation of the spectral matrix ϱ….Pages 140-149
Special properties of the spectral representation, spectral multiplicities….Pages 150-161
L 2 -solutions and essential spectrum….Pages 162-171
Differential operators with periodic coefficients….Pages 172-190
Appendix to section 12: Operators with periodic coefficients on the half-line….Pages 191-193
Oscillation theory for regular Sturm-Liouville operators….Pages 194-212
Oscillation theory for singular Sturm-Liouville operators….Pages 213-226
Essential spectrum and absolutely continuous spectrum of Sturm-Liouville operators….Pages 227-241
Oscillation theory for Dirac systems, essential spectrum and absolutely continuous spectrum….Pages 242-255
Some explicitly solvable problems….Pages 256-294