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Hardcover ISBN:  9781470448905 
Product Code:  MMONO/247 
List Price:  $165.00 
MAA Member Price:  $148.50 
AMS Member Price:  $132.00 
eBook ISBN:  9781470450236 
Product Code:  MMONO/247.E 
List Price:  $155.00 
MAA Member Price:  $139.50 
AMS Member Price:  $124.00 
Hardcover ISBN:  9781470448905 
eBook ISBN:  9781470450236 
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 DetailsTranslations of Mathematical MonographsVolume: 247; 2018; 158 ppMSC: 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 finitedifference 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.
ReadershipGraduate 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

SchmidtSonin 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


Additional Material

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


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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 finitedifference 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.
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

SchmidtSonin 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