Spectral Theory of Canonical Differential Systems. Method of Operator Identities

The spectral theory of ordinary differential operators L and of the equations (0.1) Ly= AY connected with such operators plays an important role in a number of problems both in physics and in mathematics. Let us give some examples of differential operators and equations, the spectral theory of which is well developed. Example 1. The Sturm-Liouville operator has the form (see [6]) 2 d y (0.2) Ly = - dx + u(x)y = Ay. 2 In quantum mechanics the Sturm-Liouville operator L is known as the one-dimen sional Schrodinger operator. The behaviour of a quantum particle is described in terms of spectral characteristics of the operator L. Example 2. The vibrations of a nonhomogeneous string are described by the equa tion (see [59]) p(x) ~ o. (0.3) The first results connected with equation (0.3) were obtained by D. Bernoulli and L. Euler. The investigation of this equation and of its various generalizations continues to be a very active field (see, e.g., [18], [19]). The spectral theory of the equation (0.3) has also found important applications in probability theory [20]. Example 3. Dirac-type systems of the form (0.4) } where a(x) = a(x), b(x) = b(x), are also well studied. Among the works devoted to the spectral theory of the system (0.4) the well-known article of M. G. KreIn [48] deserves special mention.

Inhalt
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 Livic-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.