## Spectral Theory of Ordinary Differential OperatorsThese 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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### Contents

Introduction | 1 |

Fundamental properties and general assumptions | 23 |

Proof of the Lagrange identity for n 2 | 35 |

Copyright | |

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absolute value absolutely continuous analytic Applications arbitrary assume boundary conditions bounded calculation choose close coefficients complete consider constant corresponding D(To deficiency indices defined determined differential equation differential expression domain Edited eigenfunction eigenvalues equal equivalent Example exists finite follows formula functions fundamental system give given hand Hence holds implies initial integrable interval least lie right lies left limit linear linearly independent matrix maximal means measurable multiplicity notice Obviously periodic positive problem Proceedings Proof proof of Theorem properties prove regular resolvent respect restriction result satisfies the boundary self-adjoint extension self-adjoint realization separated boundary conditions solution space spectral spectral representation Sturm-Liouville operators sufficiently symmetric term Theorem theory tion uniquely zeros in a,b