# Spectral Theory of Canonical Differential Systems. Method of Operator Identities

Book

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

1. Front Matter
Pages i-vi
2. Lev A. Sakhnovich
Pages 1-13
3. Lev A. Sakhnovich
Pages 15-27
4. Lev A. Sakhnovich
Pages 29-37
5. Lev A. Sakhnovich
Pages 39-48
6. Lev A. Sakhnovich
Pages 49-65
7. Lev A. Sakhnovich
Pages 67-76
8. Lev A. Sakhnovich
Pages 77-93
9. Lev A. Sakhnovich
Pages 95-106
10. Lev A. Sakhnovich
Pages 107-116
11. Lev A. Sakhnovich
Pages 117-129
12. Lev A. Sakhnovich
Pages 131-151
13. Lev A. Sakhnovich
Pages 153-166
14. Lev A. Sakhnovich
Pages 167-191
15. Back Matter
Pages 193-202

### 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

1. 1.Department of MathematicsAcademy of CommunicationOdessaUkraine

### Bibliographic information

• Book Title Spectral Theory of Canonical Differential Systems. Method of Operator Identities
• Authors L.A. Sakhnovich
• Series Title Operator Theory: Advances and Applications
• DOI https://doi.org/10.1007/978-3-0348-8713-7
• Copyright Information Birkhäuser Verlag 1999
• Publisher Name Birkhäuser, Basel
• eBook Packages
• Hardcover ISBN 978-3-7643-6057-3
• Softcover ISBN 978-3-0348-9739-6
• eBook ISBN 978-3-0348-8713-7
• Edition Number 1
• Number of Pages VI, 202
• Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site
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