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Spectral Theory of Canonical Differential Systems. Method of Operator Identities

  • Lev A. Sakhnovich

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

Table of contents

  1. Front Matter
    Pages i-vi
  2. Lev A. Sakhnovich
    Pages 1-13
  3. Lev A. Sakhnovich
    Pages 29-37
  4. Lev A. Sakhnovich
    Pages 39-48
  5. Lev A. Sakhnovich
    Pages 49-65
  6. Lev A. Sakhnovich
    Pages 67-76
  7. Lev A. Sakhnovich
    Pages 95-106
  8. Lev A. Sakhnovich
    Pages 107-116
  9. Lev A. Sakhnovich
    Pages 117-129
  10. Lev A. Sakhnovich
    Pages 131-151
  11. Lev A. Sakhnovich
    Pages 153-166
  12. Back Matter
    Pages 193-202

About this book

Introduction

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.

Keywords

differential operator mechanics operator quantum mechanics schrödinger equation spectral theory

Authors and affiliations

  • Lev A. Sakhnovich
    • 1
  1. 1.Department of MathematicsAcademy of CommunicationOdessaUkraine

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