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Spectral Theory of Canonical Differential Systems. Method of Operator Identities

  • Book
  • © 1999

Overview

Authors:
  1. Lev A. Sakhnovich

    1. Department of Mathematics, Academy of Communication, Odessa, Ukraine

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Part of the book series: Operator Theory: Advances and Applications (OT, volume 107)

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-vi
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  2. Introduction

    • Lev A. Sakhnovich
    Pages 1-13

  3. Factorization of Operator-valued Transfer Functions

    • Lev A. Sakhnovich
    Pages 15-27

  4. Operator Identities and S-nodes

    • Lev A. Sakhnovich
    Pages 29-37

  5. Continual Factorization

    • Lev A. Sakhnovich
    Pages 39-48

  6. Spectral Problems on the Half-line

    • Lev A. Sakhnovich
    Pages 49-65

  7. Spectral Problems on the Line

    • Lev A. Sakhnovich
    Pages 67-76

  8. Weyl-Titchmarsh Functions of Periodic Canonical Systems

    • Lev A. Sakhnovich
    Pages 77-93

  9. Division of Canonical Systems into Subclasses

    • Lev A. Sakhnovich
    Pages 95-106

  10. Uniqueness Theorems

    • Lev A. Sakhnovich
    Pages 107-116

  11. Weyl Discs and Points

    • Lev A. Sakhnovich
    Pages 117-129

  12. A Class of Canonical Systems

    • Lev A. Sakhnovich
    Pages 131-151

  13. Classical Spectral Problems

    • Lev A. Sakhnovich
    Pages 153-166

  14. Nonlinear Integrable Equations and the Method of the Inverse Spectral Problem

    • Lev A. Sakhnovich
    Pages 167-191

  15. Back Matter

    Pages 193-202
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Keywords

About this book

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.

Authors and Affiliations

  • Department of Mathematics, Academy of Communication, Odessa, Ukraine

    Lev A. Sakhnovich

Bibliographic Information

  • Book Title: Spectral Theory of Canonical Differential Systems. Method of Operator Identities

  • Authors: Lev A. Sakhnovich

  • Series Title: Operator Theory: Advances and Applications

  • DOI: https://doi.org/10.1007/978-3-0348-8713-7

  • Publisher: Birkhäuser Basel

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Basel AG 1999

  • Hardcover ISBN: 978-3-7643-6057-3Published: 01 April 1999

  • Softcover ISBN: 978-3-0348-9739-6Published: 04 October 2012

  • eBook ISBN: 978-3-0348-8713-7Published: 06 December 2012

  • Series ISSN: 0255-0156

  • Series E-ISSN: 2296-4878

  • Edition Number: 1

  • Number of Pages: VI, 202

  • Topics: Analysis

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