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Inverse Problems in the Theory of Small Oscillations
 
Vladimir Marchenko National Academy of Sciences of Ukraine, Kharkiv, Ukraine
Victor Slavin National Academy of Sciences of Ukraine, Kharkiv, Ukraine
Inverse Problems in the Theory of Small Oscillations
Hardcover ISBN:  978-1-4704-4890-5
Product Code:  MMONO/247
List Price: $165.00
MAA Member Price: $148.50
AMS Member Price: $132.00
eBook ISBN:  978-1-4704-5023-6
Product Code:  MMONO/247.E
List Price: $155.00
MAA Member Price: $139.50
AMS Member Price: $124.00
Hardcover ISBN:  978-1-4704-4890-5
eBook: ISBN:  978-1-4704-5023-6
Product Code:  MMONO/247.B
List Price: $320.00 $242.50
MAA Member Price: $288.00 $218.25
AMS Member Price: $256.00 $194.00
Inverse Problems in the Theory of Small Oscillations
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Inverse Problems in the Theory of Small Oscillations
Vladimir Marchenko National Academy of Sciences of Ukraine, Kharkiv, Ukraine
Victor Slavin National Academy of Sciences of Ukraine, Kharkiv, Ukraine
Hardcover ISBN:  978-1-4704-4890-5
Product Code:  MMONO/247
List Price: $165.00
MAA Member Price: $148.50
AMS Member Price: $132.00
eBook ISBN:  978-1-4704-5023-6
Product Code:  MMONO/247.E
List Price: $155.00
MAA Member Price: $139.50
AMS Member Price: $124.00
Hardcover ISBN:  978-1-4704-4890-5
eBook ISBN:  978-1-4704-5023-6
Product Code:  MMONO/247.B
List Price: $320.00 $242.50
MAA Member Price: $288.00 $218.25
AMS Member Price: $256.00 $194.00
  • Book Details
     
     
    Translations of Mathematical Monographs
    Volume: 2472018; 158 pp
    MSC: Primary 70; 65; 35

    Inverse problems of spectral analysis deal with the reconstruction of operators of the specified form in Hilbert or Banach spaces from certain of their spectral characteristics. An interest in spectral problems was initially inspired by quantum mechanics. The main inverse spectral problems have been solved already for Schrödinger operators and for their finite-difference analogues, Jacobi matrices.

    This book treats inverse problems in the theory of small oscillations of systems with finitely many degrees of freedom, which requires finding the potential energy of a system from the observations of its oscillations. Since oscillations are small, the potential energy is given by a positive definite quadratic form whose matrix is called the matrix of potential energy. Hence, the problem is to find a matrix belonging to the class of all positive definite matrices. This is the main difference between inverse problems studied in this book and the inverse problems for discrete analogues of the Schrödinger operators, where only the class of tridiagonal Hermitian matrices are considered.

    Readership

    Graduate students and researchers interested in inverse problems and scattering theory.

  • Table of Contents
     
     
    • Chapters
    • Direct problem of the oscillation theory of loaded strings
    • Eigenvectors of tridiagonal Hermitian matrices
    • Spectral function of tridiagonal Hermitian matrix
    • Schmidt-Sonin orthogonalization process
    • Construction of the tridiagonal matrix by given spectral functions
    • Reconstruction of tridiagonal matrices by two spectra
    • Solution methods for inverse problems
    • Small oscillations, potential energy matrix and $\mathbf {L}$-matrix, direct and inverse problems of the theory of small oscillations
    • Observable and computable values. Reducing inverse problems of the theory of small oscillations to the inverse problem of spectral analysis for Hermitian matrices
    • General solution for the inverse problem of spectral analysis for Hermitian matrices
    • Interaction of particles and the systems with pairwise interactions
    • Indecomposable systems, $\mathbf {M}$-extensions and the graph of interactions
    • The main lemma
    • Reconstructing a Hermitian matrix $\textbf {M}\in \mathfrak {M}(m)$ using its spectral data, restricted to a completely $\textbf {M}$-extendable set
    • Properties of completely $\textbf {M}$-extendable sets
    • Examples of $\textbf {L}$-extendable subsets
    • Computing masses of particles using the $\textbf {L}$-matrix of a system
    • Reconstructing a Hermitian matrix using its spectrum and spectra of several its perturbations
    • The inverse scattering problem
    • Solving the inverse problem of the theory of small oscillations numerically
    • Analysis of spectra for the discrete Fourier transform
    • Computing the coordinates of eigenvectors of an $\textbf {L}$-matrix, corresponding to observable particles
    • A numerical orthogonalization method for a set of vectors
    • A recursion for computing the coordinates for eigenvectors of an $\textbf {L}$-matrix
    • Examples of solving numerically the inverse problem of the theory of small oscillations
  • Reviews
     
     
    • The books of Vladimir Marchenko and Viktor Slavin have long been classics of the spectral theory of differential operators. More than one generation of specialists have been brought up on their books. This translation will certainly make a significant contribution to the development of spectral theory.

      Azamat M. Akhtyamov, Mathematical Reviews
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2472018; 158 pp
MSC: Primary 70; 65; 35

Inverse problems of spectral analysis deal with the reconstruction of operators of the specified form in Hilbert or Banach spaces from certain of their spectral characteristics. An interest in spectral problems was initially inspired by quantum mechanics. The main inverse spectral problems have been solved already for Schrödinger operators and for their finite-difference analogues, Jacobi matrices.

This book treats inverse problems in the theory of small oscillations of systems with finitely many degrees of freedom, which requires finding the potential energy of a system from the observations of its oscillations. Since oscillations are small, the potential energy is given by a positive definite quadratic form whose matrix is called the matrix of potential energy. Hence, the problem is to find a matrix belonging to the class of all positive definite matrices. This is the main difference between inverse problems studied in this book and the inverse problems for discrete analogues of the Schrödinger operators, where only the class of tridiagonal Hermitian matrices are considered.

Readership

Graduate students and researchers interested in inverse problems and scattering theory.

  • Chapters
  • Direct problem of the oscillation theory of loaded strings
  • Eigenvectors of tridiagonal Hermitian matrices
  • Spectral function of tridiagonal Hermitian matrix
  • Schmidt-Sonin orthogonalization process
  • Construction of the tridiagonal matrix by given spectral functions
  • Reconstruction of tridiagonal matrices by two spectra
  • Solution methods for inverse problems
  • Small oscillations, potential energy matrix and $\mathbf {L}$-matrix, direct and inverse problems of the theory of small oscillations
  • Observable and computable values. Reducing inverse problems of the theory of small oscillations to the inverse problem of spectral analysis for Hermitian matrices
  • General solution for the inverse problem of spectral analysis for Hermitian matrices
  • Interaction of particles and the systems with pairwise interactions
  • Indecomposable systems, $\mathbf {M}$-extensions and the graph of interactions
  • The main lemma
  • Reconstructing a Hermitian matrix $\textbf {M}\in \mathfrak {M}(m)$ using its spectral data, restricted to a completely $\textbf {M}$-extendable set
  • Properties of completely $\textbf {M}$-extendable sets
  • Examples of $\textbf {L}$-extendable subsets
  • Computing masses of particles using the $\textbf {L}$-matrix of a system
  • Reconstructing a Hermitian matrix using its spectrum and spectra of several its perturbations
  • The inverse scattering problem
  • Solving the inverse problem of the theory of small oscillations numerically
  • Analysis of spectra for the discrete Fourier transform
  • Computing the coordinates of eigenvectors of an $\textbf {L}$-matrix, corresponding to observable particles
  • A numerical orthogonalization method for a set of vectors
  • A recursion for computing the coordinates for eigenvectors of an $\textbf {L}$-matrix
  • Examples of solving numerically the inverse problem of the theory of small oscillations
  • The books of Vladimir Marchenko and Viktor Slavin have long been classics of the spectral theory of differential operators. More than one generation of specialists have been brought up on their books. This translation will certainly make a significant contribution to the development of spectral theory.

    Azamat M. Akhtyamov, Mathematical Reviews
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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