Spectral Theory of Canonical Differential Systems. Method of Operator Identities

1 Factorization of Operator-valued Transfer Functions.- 1.1 Realization of operator-valued functions.- 1.2 A factorization method.- 1.3 Factorization of rational operator-valued functions.- 2 Operator Identities and S-Nodes.- 2.1 Elementary properties of S-nodes.- 2.2 Symmetric S-nodes.- 2.3 Inherited properties of factors.- 3 Continual Factorization.- 3.1 The main continual factorization theorem.- 3.2 Bounded operator-valued functions.- 4 Spectral Problems on the Half-line.- 4.1 Basic notions of spectral theory.- 4.2 Direct and inverse spectral problems.- 4.3 Livšic-Brodski? nodes and the spectral theory of canonical systems.- 5 Spectral Problems on the Line.- 5.1 Spectral data of a canonical system.- 5.2 Spectral problems and S-nodes.- 5.3 The inverse spectral problem.- 6 Weyl-Titchmarsh Functions of Periodic Canonical Systems.- 6.1 Multipliers and their behavior.- 6.2 Weyl-Titchmarsh functions.- 6.3 Singular points of the Weyl-Titchmarsh matrix function.- 7 Division of Canonical Systems into Subclasses.- 7.1 An effective solution of the inverse problem.- 7.2 Two principles of dividing a class of canonical systems into subclasses.- 8 Uniqueness Theorems.- 8.1 Monodromy matrix and uniqueness theorems.- 8.2 Spectral data and uniqueness theorems.- 9 Weyl Discs and Points.- 9.1 Basic notions.- 9.2 Symmetric operators and deficiency indices.- 9.3 Weyl-Titchmarsh matrix functions on the line.- 9.4 Weyl-Titchmarsh matrix function of a system with shifted argument.- 10 A Class of Canonical Systems.- 10.1 Asymptotic formulas.- 10.2 Spectral analysis.- 10.3 Transformed canonical systems.- 10.4 Dirac-type systems.- 10.5 An inverse problem.- 10.6 On the limit Titchmarsh-Weyl function.- 11 Classical Spectral Problems.- 11.1 Generalized string equation (direct spectral problem).- 11.2 Matrix Sturm-Liouville equation (direct spectral problem).- 11.3 Inverse spectral problem.- 12 Nonlinear Integrable Equations and the Method of the Inverse Spectral Problem.- 12.1 Evolution of the spectral data.- 12.2 Some classical nonlinear equations.- 12.3 On the unique solvability of the mixed problem.- 12.4 A hierarchy of nonlinear equations and asymptotic behavior of Weyl-Titchmarsh functions.- Comments.- References.