Contents -- Foreword -- Preface -- Acknowledgments -- Introduction -- 1. Relation Between Spectral and Oscillatory Properties for the Matrix Sturm-Liouville Problem -- 1.1 Problem on a Finite Interval -- 1.2 Problem on the Half-Line -- 1.3 Bibliographical Comments -- 1.3.1 An Example of the Operator of (1.1)-(1.3) Type with the Purely Absolutely Continuous Spectrum -- 1.3.2 Oscillatory Theory of Partial Differential Equations -- 1.3.3 Bochner Integral -- 2. Fundamental System of Solutions for an Operator Differential Equation with a Singular Boundary Condition -- 2.1 Separated Self-Adjoint Boundary Conditions -- 2.2 A Construction for the Fundamental Solution of the Boundary-Value Problem -- 2.3 Self-consistency of the Fundamental Solution of the Self- Adjoint Problem and the Evolution of the Corresponding Hermitian Relation (or Lagrangian Plane in H @ H ) -- 2.4 A Different Construction for the Fundamental Solution of the Boundary-Value Problem -- 2.5 Bibliographical Comments -- 3. Dependence of the Spectrum of Operator Boundary Problems on Variations of a Finite or Semi-Infinite Interval -- 3.1 Dependence of Eigenvalues and the Greatest Lower Bound of the Spectrum for a Semi-Bounded Below Differential Operator on Variations of the Interval -- 3.2 Continuity and Monotonicity of the Greatest Lower Bound of the Essential Spectrum for Semi-Bounded Below Differential Operators -- 3.3 Bibliographical Comments -- 4. Relation Between Spectral and Oscillatory Properties for Operator Differential Equations of Arbitrary Order -- 4.1 Oscillatory Theorem for Operator Differential Equations of Even Order -- 4.2 Comparison and Alternation Theorems -- 4.3 Multiplicative Representation of Positive Differential Operators and Its Applications -- 4.4 Discrete Levels in the Gaps of the Essential Spectrum -- 4.5 Bibliographical Comments.

4.5.1 Symplectic Interpretation of Sturm-Type Theorems and Their Operator-Theoretical Proofs -- 5. Self-Adjoint Extensions of Systems of Differential Equations of Arbitrary Order on an Infinite Interval in the Absolutely Indefinite Case -- 5.1 A Description of Self-Adjoint Extensions of Differential Operators of Arbitrary Order with Operator-Valued Coefficients on an Infinite Interval -- 5.2 Parametrization of the Characteristic Operator Function -- 5.3 Bibliographical Comments -- 5.3.1 Everitt-Zettl Problem, Brusentsev's Example and Two Open Questions -- 5.3.2 On the Deficiency Indices of Scalar Symmetric Dif- ferential Operators of General Kind on the Half-Axis (Solved and Unsolved Questions) -- 5.3.3 Kogan-Rofe-Beketov's Asymptotic Theorems and Deficiency Indices of Symmetric Differential Operator -- 5.3.4 R. C. Gilbert's Class of Formally Self-Adjoint Ordi- nary Differential Operators Whose Deficiency Numbers Differ by an Arbitrary Pre-Assigned Positive Integer -- 5.3.5 Deficiency Indices of Symmetric Differential Systems of the First and Arbitrary Orders and Some Open Questions -- 5.3.6 Some Comments on Hilbert's 21-st Problem and Bolibrukh Counterexample -- 5.3.7 Some Comments on Section 5.2 -- 5.3.8 Characteristic Properties of Weyl Solutions for the Sturm-Liouville and Dirac Equations. V. A. Marchenko's Theorems -- 6. Discrete Levels in Spectral Gaps of Perturbed Schrodinger and Hill Operators -- 6.1 Factorized Phase Matrix of the Perturbed System and the Discrete Spectrum in Gaps of the Continuous Spectrum -- 6.2 Generalization of D' Alembert-Liouville-Ostrogradsky For- mula and Its Application to Growth Estimates for Solutions of Canonical Almost Periodic Systems.

6.3 A Reduction of the Problem about the Number of Discrete Levels in a Finite Spectral Gap of One-Dimensional Schrodinger Operator to a Similar Problem for a Semi- Infinite Gap of the Transformed Schrodinger Operator -- 6.4 Applications to Perturbations of Hill Operators and Schrodinger Operators with an Almost Periodic Potential. Kneser-Type Constants and Effective Masses -- 6.5 Discrete Levels in Spectral Gaps of Perturbed Hill Operators -- 6.6 Bibliographical Comments -- 6.6.1 Some Comments on Section 6.1 -- 6.6.2 Some Comments on Section 6.2 -- 6.6.3 Some Comments on Section 6.3 and Differential Inequalities -- 6.6.4 The Spectrum of Self-Adjoint Periodic and Almost Periodic Operators. Marchenko-Ostrovskiy and Pastur-Tkachenko Theorems -- 6.6.5 The Spectrum of Non-Self-Adjoint Operators. Batchenko-Gesztesy and Rofe-Beketov Theorems -- 6.6.6 The Spectrum of Perturbed Self-Adjoint and Non-Self-Adjoint Operators -- 6.6.7 Matrix Operators or Differential-Algebraic Operators and Related Topics -- 6.6.8 The Inverse Sturm-Liouville Problem for the Spectral Matrix on the Whole Axis -- 6.6.9 Three Open Questions -- Appendix A Self-Adjoint Extensions of Differential Opera- tors on a Finite Interval in Spaces of Vector-Functions -- A.1 Hermitian Relations -- A.1.1 Canonical Form of Hermitian Relations -- A.1.2 Various Forms of Hermitian Relations. Dissipative and Sectorial Relations -- A.2 Self-Adjoint Boundary Conditions for Infinite Systems of Second Order Differential Equations -- A.3 Self-Adjoint Extensions of Differential Operators of Arbitrary Order with Operator Coefficients -- A.3.1 The Expressions of Order 2n -- A.3.2 The Expressions of Order 2n - 1 -- A.3.3 Continuous Dependence of Initial Data on Solutions of Differential Equations -- A.4 Bibliographical Comments.

A.4.1 M. L. Gorbachuk Theory of Generalized Boundary Values. Abstract Boundary Conditions -- A.4.2 Strong Resolvent Convergence of Kermitian Relations and Operators with Non-Dense Domains and Resolution of the Identity. Stability of Eigenvalues -- A.4.3 Characteristic Operator and Boundary-Value Prob- lems with Separated Boundary Conditions -- Bibliography -- Monographs -- List of Symbols -- Index.