Spectral Theory of Ordinary Differential Operators

1. Essential spectrum and absolutely continuous spectrum of Sturm-Liouville operators

   • Joachim Weidmann

   Pages 227-241

2. Oscillation theory for Dirac systems, essential spectrum and absolutely continuous spectrum

   • Joachim Weidmann

   Pages 242-255

3. Some explicitly solvable problems

   • Joachim Weidmann

   Pages 256-294

4. Back Matter

   Pages 295-303

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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.

• Dirac operator
• Hilbert space
• differential equation
• maximum
• minimum

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