Synopses & Reviews
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.
Table of Contents
Contents: Formally self-adjoint differential expressions.- Fundamental properties and general assumptions.- The minimal operator and the maximal operator.- Deficiency indices and self-adjoint extensions of T .- The solutions of the inhomogeneous differential equation ( - )u = f; Weyl's alternative.- Limit point - limit circle criteria.- The resolvent of self-adjoint extensions of T .- The spectral representation of self-adjoint extensions of T .- Computation of the spectral matrix .- Special properties of the spectral representation, spectral multiplicities.- L -Solutions and essential spectrum.- Differential operators with periodic coefficients.- 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.- References.- Subject index.