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Bitangential Direct and Inverse Problems for Systems of Integral and Differential Equations

Book Description

This largely self-contained treatment surveys, unites and extends some 20 years of research on direct and inverse problems for canonical systems of integral and differential equations and related systems. Five basic inverse problems are studied in which the main part of the given data is either a monodromy matrix; an input scattering matrix; an input impedance matrix; a matrix valued spectral function; or an asymptotic scattering matrix. The corresponding direct problems are also treated. The book incorporates introductions to the theory of matrix valued entire functions, reproducing kernel Hilbert spaces of vector valued entire functions (with special attention to two important spaces introduced by L. de Branges), the theory of J-inner matrix valued functions and their application to bitangential interpolation and extension problems, which can be used independently for courses and seminars in analysis or for self-study. A number of examples are presented to illustrate the theory.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright Page
  4. Dedication
  5. Contents
  6. Preface
  7. 1: Introduction
    1. 1.1 The matrizant as a chain
    2. 1.2 Monodromy matrices of regular systems
    3. 1.3 Canonical integral systems
    4. 1.4 Singular, right regular and right strongly regular matrizants
    5. 1.5 Input scattering matrices
    6. 1.6 Chains of associated pairs of the first kind
    7. 1.7 The bitangential direct input scattering problem
    8. 1.8 Bitangential inverse monodromy and inverse scattering problems
    9. 1.9 The generalized Schur interpolation problem
    10. 1.10 Identifying matrizants
    11. 1.11 Input impedance matrices and spectral functions
    12. 1.12 de Branges spaces
    13. 1.13 Bitangential direct and inverse input impedance and spectral problems
    14. 1.14 Krein extension problems and Dirac systems
    15. 1.15 Direct and inverse problems for Dirac Krein systems
    16. 1.16 Supplementary notes
  8. 2: Canonical systems and related differential equations
    1. 2.1 Canonical integral systems
    2. 2.2 Connections with canonical differential systems
    3. 2.3 The matrizant and its properties
    4. 2.4 Regular case: Monodromy matrix
    5. 2.5 Multiplicative integral formulas
    6. 2.6 The Feller Krein string equation
    7. 2.7 Differential systems with potential
    8. 2.8 Dirac Krein systems
    9. 2.9 The Schrodinger equation
    10. 2.10 Supplementary notes
  9. 3: Matrix valued functions in the Nevanlinna class
    1. 3.1 Preliminaries on the Nevanlinna class
    2. 3.2 Linear fractional transformations and Redheffer transformations
    3. 3.3 The Riesz Herglotz Nevanlinna representation
    4. 3.4 The class entire mvfs
    5. 3.5 The class Pi
    6. 3.6 Fourier transforms and Paley Wiener theorems
    7. 3.7 Entire inner mvfs
    8. 3.8 J contractive, J inner and entire J inner mvfs
    9. 3.9 Associated pairs of the first kind
    10. 3.10 Singular and right and left regular J inner mvfs
    11. 3.11 Linear fractional transformations of S
    12. 3.12 Linear fractional transformations in C
    13. 3.13 Associated pairs of the second kind
    14. 3.14 Supplementary notes
  10. 4: Interpolation problems, resolvent matrices and de Branges spaces
    1. 4.1 The Nehari problem
    2. 4.2 The generalized Schur interpolation problem
    3. 4.3 Right and left strongly regular J inner mvfs
    4. 4.4 The generalized Caratheodory interpolation problem
    5. 4.5 Detour on scalar determinate interpolation problems
    6. 4.6 The reproducing kernel Hilbert space
    7. 4.7 de Branges inclusion theorems
    8. 4.8 A description of
    9. 4.9 The classes A regular and B regular J inner mvfs
    10. 4.10 de Branges matrices and de Branges spaces
    11. 4.11 A coisometry from
    12. 4.12 Formulas for resolvent matrices
    13. 4.13 Formulas for resolvent matrices
    14. 4.14 Supplementary notes
  11. 5: Chains that are matrizants and chains of associated pairs
    1. 5.1 Continuous chains of entire J inner mvfs
    2. 5.2 Chains that are matrizants
    3. 5.3 Continuity of chains of associated pairs
    4. 5.4 Type functions for chains
    5. 5.5 Supplementary notes
  12. 6: The bitangential direct input scattering problem
    1. 6.1 The set of input scattering matrices
    2. 6.2 Parametrization
    3. 6.3 Regular canonical integral systems
    4. 6.4 Limit balls for input scattering matrices
    5. 6.5 The full rank case
    6. 6.6 Rank formulas
    7. 6.7 Regular systems
    8. 6.8 The limit point case
    9. 6.9 The diagonal case
    10. 6.10 A Weyl Titchmarsh
    11. 6.11 Supplementary notes
  13. 7: Bitangential direct input impedance and spectral problems
    1. 7.1 Input impedance matrices
    2. 7.2 Limit balls for input impedance matrices
    3. 7.3 Formulas for the ranks of semiradii of the limit ball
    4. 7.4 Bounded mass functions and full rank end points
    5. 7.5 The limit point case
    6. 7.6 The Weyl Titchmarsh characterization of the input impedance
    7. 7.7 Spectral functions for canonical systems
    8. 7.8 Parametrization of the set image
    9. 7.9 Parametrization of the set image for regular canonical integral systems
    10. 7.10 Pseudospectral and spectral functions for singular systems
    11. 7.11 Supplementary notes
  14. 8: Inverse monodromy problems
    1. 8.1 Some simple illustrative examples
    2. 8.2 Extremal solutions
    3. 8.3 Solutions for image
    4. 8.4 Connections with the Livsic model of a Volterra node
    5. 8.5 Conditions for the uniqueness of normalized Hamiltonians
    6. 8.6 Solutions with symplectic and or real matrizants
    7. 8.7 Entire homogeneous resolvent matrices
    8. 8.8 Solutions with homogeneous matrizants
    9. 8.9 Extremal solutions for
    10. 8.10 The unicellular case for
    11. 8.11 Solutions with symmetric type
    12. 8.12 The inverse monodromy problem for
    13. 8.13 Examples of Hamiltonians with constant determinant
    14. 8.14 Supplementary notes
  15. 9: Bitangential Krein extension problems
    1. 9.1 Helical extension problems
    2. 9.2 Bitangential helical extension problems
    3. 9.3 The Krein accelerant extension problem
    4. 9.4 Continuous analogs of the Schur extension problem
    5. 9.5 A bitangential generalization of the Schur extension problem
    6. 9.6 The Nehari extension problem for mvfs in Wiener class
    7. 9.7 Continuous analogs of the Schur extension problem for mvfs in the Wiener class
    8. 9.8 Bitangential Schur extension problems in the Wiener class
    9. 9.9 Supplementary notes
  16. 10: Bitangential inverse input scattering problems
    1. 10.1 Existence and uniqueness of solutions
    2. 10.2 Formulas for the solution of the inverse input scattering problem
    3. 10.3 Input scattering matrices in the Wiener class
    4. 10.4 Examples with diagonal mvfs image and image
    5. 10.5 Supplementary notes
  17. 11: Bitangential inverse input impedance and spectral problems
    1. 11.1 Existence and uniqueness of solutions
    2. 11.2 Formulas for the solutions
    3. 11.3 Input impedance matrices in the Wiener class
    4. 11.4 Examples with diagonal mvfs image and image
    5. 11.5 The bitangential inverse spectral problem
    6. 11.6 An example
    7. 11.7 Supplementary notes
  18. 12: Direct and inverse problems for Dirac Krein systems
    1. 12.1 Factoring Hamiltonians corresponding to DK systems
    2. 12.2 Matrizants of canonical differential systems corresponding to DK systems
    3. 12.3 Direct and inverse monodromy problems for DK systems
    4. 12.4 Direct and inverse input scattering problems for DK systems
    5. 12.5 Direct and inverse input impedance problems for DK systems
    6. 12.6 Direct and inverse spectral problems for DK systems
    7. 12.7 The Krein algorithms for the inverse input scattering and impedance problems
    8. 12.8 The left transform image for image
    9. 12.9 Asymptotic equivalence matrices
    10. 12.10 Asymptotic scattering matrices S matrices
    11. 12.11 The inverse asymptotic scattering problem
    12. 12.12 More on spectral functions of DK systems
    13. 12.13 Supplementary notes
  19. References
  20. Symbol index
  21. Index