Complete Second Order Linear Differential Equations in Hilbert Spaces

Incomplete second order linear differential equations in Banach spaces as well as first order equations have become a classical part of functional analysis. This monograph is an attempt to present a unified systematic theory of second order equations y" (t) + Ay' (t) + By (t) = 0 including well-posedness of the Cauchy problem as well as the Dirichlet and Neumann problems. Exhaustive yet clear answers to all posed questions are given. Special emphasis is placed on new surprising effects arising for complete second order equations which do not take place for first order and incomplete second order equations. For this purpose, some new results in the spectral theory of pairs of operators and the boundary behavior of integral transforms have been developed. The book serves as a self-contained introductory course and a reference book on this subject for undergraduate and post- graduate students and research mathematicians in analysis. Moreover, users will welcome having a comprehensive study of the equations at hand, and it gives insight into the theory of complete second order linear differential equations in a general context - a theory which is far from being fully understood.

I. Well-posedness of boundary-value problems.- to Part I.- 1. Joint spectrum of commuting normal operators and its position. Estimates for roots of second order polynomials. Definition of well-posedness of boundary-value problems.- 2. Well-posedness of boundary-value problems for equation (1) in the case of commuting self-adjoint A and B.- 3. The Cauchy problem.- 4. Boundary-value problems on a finite segment.- II. Initial data of solutions.- to Part II.- 5. Boundary behaviour of an integral transform R(t) as t ? 0 depending on the sub-integral measure.- 6. Initial data of solutions.- III. Extension, stability, and stabilization of weak solutions.- to Part III.- 7. The general form of weak solutions.- 8. Fatou-Riesz property.- 9. Extension of weak solutions.- 10. Stability and stabilization of weak solutions.- IV. Boundary-value problems on a half-line.- to Part IV.- 11. The Dirichlet problem on a half-line.- 12. The Neumann problem on a half-line.- Commentaries on the literature.- List of symbols.