Spectral Theory of Operator Pencils, Hermite-Biehler Functions, and their Applications

  • Manfred Möller
  • Vyacheslav Pivovarchik

Part of the Operator Theory: Advances and Applications book series (OT, volume 246)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. Operator Pencils

    1. Front Matter
      Pages 1-1
    2. Manfred Möller, Vyacheslav Pivovarchik
      Pages 3-31
    3. Manfred Möller, Vyacheslav Pivovarchik
      Pages 33-67
    4. Manfred Möller, Vyacheslav Pivovarchik
      Pages 69-82
    5. Manfred Möller, Vyacheslav Pivovarchik
      Pages 83-115
  3. Hermite–Biehler Functions

    1. Front Matter
      Pages 117-117
    2. Manfred Möller, Vyacheslav Pivovarchik
      Pages 119-152
    3. Manfred Möller, Vyacheslav Pivovarchik
      Pages 153-173
  4. Direct and Inverse Problems

    1. Front Matter
      Pages 175-175
    2. Manfred Möller, Vyacheslav Pivovarchik
      Pages 177-214
    3. Manfred Möller, Vyacheslav Pivovarchik
      Pages 215-248
  5. Background Material

    1. Front Matter
      Pages 249-249
    2. Manfred Möller, Vyacheslav Pivovarchik
      Pages 251-268
    3. Manfred Möller, Vyacheslav Pivovarchik
      Pages 269-283
    4. Manfred Möller, Vyacheslav Pivovarchik
      Pages 285-344
    5. Manfred Möller, Vyacheslav Pivovarchik
      Pages 345-387
  6. Back Matter
    Pages 389-412

About this book

Introduction

The theoretical part of this monograph examines the distribution of the spectrum of operator polynomials, focusing on quadratic operator polynomials with discrete spectra. The second part is devoted to applications. Standard spectral problems in Hilbert spaces are of the form A-λI for an operator A, and self-adjoint operators are of particular interest and importance, both theoretically and in terms of applications. A characteristic feature of self-adjoint operators is that their spectra are real, and many spectral problems in theoretical physics and engineering can be described by using them. However, a large class of problems, in particular vibration problems with boundary conditions depending on the spectral parameter, are represented by operator polynomials that are quadratic in the eigenvalue parameter and whose coefficients are self-adjoint operators. The spectra of such operator polynomials are in general no more real, but still exhibit certain patterns. The distribution of these spectra is the main focus of the present volume. For some classes of quadratic operator polynomials, inverse problems are also considered. The connection between the spectra of such quadratic operator polynomials and generalized Hermite-Biehler functions is discussed in detail.

Many applications are thoroughly investigated, such as the Regge problem and damped vibrations of smooth strings, Stieltjes strings, beams, star graphs of strings and quantum graphs. Some chapters summarize advanced background material, which is supplemented with detailed proofs. With regard to the reader’s background knowledge, only the basic properties of operators in Hilbert spaces and well-known results from complex analysis are assumed.

Keywords

damped vibrations generalized Hermite-Biehler functions inverse problems operator pencils spectral asymptotics

Authors and affiliations

  • Manfred Möller
    • 1
  • Vyacheslav Pivovarchik
    • 2
  1. 1.John Knopfmacher Center for Applicable Analysis and Number TheoryUniversity of the Witwatersrand, School of MathematicsJohannesburgSouth Africa
  2. 2.Department of Algebra and GeometrySouth Ukrainian National Pedagogical UniversityOdessaUkraine

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-17070-1
  • Copyright Information Springer International Publishing Switzerland 2015
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-17069-5
  • Online ISBN 978-3-319-17070-1
  • Series Print ISSN 0255-0156
  • Series Online ISSN 2296-4878
  • About this book