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Direct and Inverse Sturm-Liouville Problems

A Method of Solution

  • Book
  • © 2020

Overview

Authors:
  1. Vladislav V. Kravchenko

    1. Department of Mathematics, Cinvestav, Querétaro, Mexico

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  • Explains in detail direct and simple methods for solving direct and inverse Sturm-Liouville and scattering problems on finite and infinite intervals
  • Includes a brief introduction to the notion and properties of transmutation operators
  • Formulates some of the most typical direct and inverse spectral and scattering problems

Part of the book series: Frontiers in Mathematics (FM)

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Table of contents (16 chapters)

  1. Front Matter

    Pages i-xi
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  2. Introduction

    • Vladislav V. Kravchenko
    Pages 1-4

  3. Typical Problem Statements

    1. Front Matter

      Pages 5-5
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    2. Preliminaries on Sturm-Liouville Equations

      • Vladislav V. Kravchenko
      Pages 7-13

    3. Direct and Inverse Sturm-Liouville Problems on Finite Intervals

      • Vladislav V. Kravchenko
      Pages 15-18

    4. Direct and Inverse Sturm-Liouville Problems on a Half-Line

      • Vladislav V. Kravchenko
      Pages 19-22

    5. Quantum Scattering Problem on the Half-Line

      • Vladislav V. Kravchenko
      Pages 23-24

    6. Scattering Problem on the Line

      • Vladislav V. Kravchenko
      Pages 25-28

    7. Inverse Scattering Transform Method

      • Vladislav V. Kravchenko
      Pages 29-31

  4. Transmutation Operators and Series Representations for Solutions of Sturm-Liouville Equations

    1. Front Matter

      Pages 33-33
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    2. Main Transmutation Operators

      • Vladislav V. Kravchenko
      Pages 35-40

    3. Construction of Transmutations and Series Representations for Solutions

      • Vladislav V. Kravchenko
      Pages 41-60

    4. Series Representations for the Kernel A(x, t) and for the Jost Solution

      • Vladislav V. Kravchenko
      Pages 61-74

  5. Solution of Direct Sturm-Liouville Problems

    1. Front Matter

      Pages 75-75
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    2. Sturm–Liouville Problems on Finite Intervals

      • Vladislav V. Kravchenko
      Pages 77-87

    3. Spectral Problems on Infinite Intervals

      • Vladislav V. Kravchenko
      Pages 89-98

  6. Solution of Inverse Sturm-Liouville Problems

    1. Front Matter

      Pages 99-99
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    2. The Inverse Sturm–Liouville Problem on a Finite Interval

      • Vladislav V. Kravchenko
      Pages 101-116

    3. Solution of the Inverse Problem on the Half-Line

      • Vladislav V. Kravchenko
      Pages 117-123

    4. Solution of the Inverse Quantum Scattering Problem on the Half-Line

      • Vladislav V. Kravchenko
      Pages 125-130
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Keywords

About this book

This book provides an introduction to the most recent developments in the theory and practice of direct and inverse Sturm-Liouville problems on finite and infinite intervals. A universal approach for practical solving of direct and inverse spectral and scattering problems is presented, based on the notion of transmutation (transformation) operators and their efficient construction. Analytical representations for solutions of Sturm-Liouville equations as well as for the integral kernels of the transmutation operators are derived in the form of functional series revealing interesting special features and lending themselves to direct and simple numerical solution of a wide variety of problems.

The book is written for undergraduate and graduate students, as well as for mathematicians, physicists and engineers interested in direct and inverse spectral problems.

Authors and Affiliations

  • Department of Mathematics, Cinvestav, QuerĂ©taro, Mexico

    Vladislav V. Kravchenko

Bibliographic Information

  • Book Title: Direct and Inverse Sturm-Liouville Problems

  • Book Subtitle: A Method of Solution

  • Authors: Vladislav V. Kravchenko

  • Series Title: Frontiers in Mathematics

  • DOI: https://doi.org/10.1007/978-3-030-47849-0

  • Publisher: Birkhäuser Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020

  • Softcover ISBN: 978-3-030-47848-3Published: 29 July 2020

  • eBook ISBN: 978-3-030-47849-0Published: 28 July 2020

  • Series ISSN: 1660-8046

  • Series E-ISSN: 1660-8054

  • Edition Number: 1

  • Number of Pages: XI, 154

  • Number of Illustrations: 1 b/w illustrations, 27 illustrations in colour

  • Topics: Analysis, Operator Theory

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