This is the first monograph devoted to the Sturm oscillatory theory for infinite systems of differential equations and its relations with the spectral theory. It aims to study a theory of self-adjoint problems for such systems, based on an elegant method of binary relations. Another topic investigated in the book is the behavior of discrete eigenvalues which appear in spectral gaps of the Hill operator and almost periodic Schrödinger operators due to local perturbations of the potential (e.g., modeling impurities in crystals).

The book is based on results that have not been presented in other monographs. The only prerequisites needed to read it are basics of ordinary differential equations and operator theory. It should be accessible to graduate students, though its main topics are of interest to research mathematicians working in functional analysis, differential equations and mathematical physics, as well as to physicists interested in spectral theory of differential operators.

Contents:

• Relation Between Spectral and Oscillatory Properties for the Matrix Sturm–Liouville Problem
• Fundamental System of Solutions for an Operator Differential Equation with a Singular Boundary Condition
• Dependence of the Spectrum of Operator Boundary Problems on Variations of a Finite or Semi-Infinite Interval
• Relation Between Spectral and Oscillatory Properties for Operator Differential Equations of Arbitrary Order
• Self-Adjoint Extensions of Systems of Differential Equations of Arbitrary Order on an Infinite Interval in the Absolutely Indefinite Case
• Discrete Levels in Spectral Gaps of Perturbed Schrödinger and Hill Operators

“The appendix is very valuable and helps the reader to find an orientation in the very voluminous literature devoted to the spectral theory of differential operators ... anybody interested in the spectral theory of differential operators will find interesting information in the book, including formulation of open problems for possible investigation.”

“This book is well-written, and a list of symbols and the index prove useful. A substantial number of open questions is also included. Although addressed primarily to the research community, the book could also be used as a graduate textbooks.”