Spectral Theory of Canonical Differential Systems. Method of Operator Identities (Operator Theory: Advances and Applications, 107)

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. Front Matter....Pages i-vi Introduction....Pages 1-13 Factorization of Operator-valued Transfer Functions....Pages 15-27 Operator Identities and S-nodes....Pages 29-37 Continual Factorization....Pages 39-48 Spectral Problems on the Half-line....Pages 49-65 Spectral Problems on the Line....Pages 67-76 Weyl-Titchmarsh Functions of Periodic Canonical Systems....Pages 77-93 Division of Canonical Systems into Subclasses....Pages 95-106 Uniqueness Theorems....Pages 107-116 Weyl Discs and Points....Pages 117-129 A Class of Canonical Systems....Pages 131-151 Classical Spectral Problems....Pages 153-166 Nonlinear Integrable Equations and the Method of the Inverse Spectral Problem....Pages 167-191 Back Matter....Pages 193-202 Theorems of factorising matrix functions and the operator identity method play an essential role in this book in constructing the spectral theory (direct and inverse problems) of canonical differential systems. The general spectral theory is then applied both to classical spectral problems (Sturm-Liouville, Dirac, string equations, Krein systems) and new, important nonlinear equations such as the nonlinear Schrödinger equation, the modified Korteweg-de Vries equation and the sinh-Gordon equation. Theorems of factorising matrix functions and the operator identity method play an essential role in this book in constructing the spectral theory (direct and inverse problems) of canonical differential systems. Includes many varied applications of the general theory.