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Inverse Problems in the Theory of Small Oscillations

About this Title

Vladimir Marchenko, National Academy of Sciences of Ukraine, Kharkiv, Ukraine and Victor Slavin, National Academy of Sciences of Ukraine, Kharkiv, Ukraine

Publication: Translations of Mathematical Monographs
Publication Year: 2018; Volume 247
ISBNs: 978-1-4704-4890-5 (print); 978-1-4704-5023-6 (online)
DOI: https://doi.org/10.1090/mmono/247
MathSciNet review: MR3839313
MSC: Primary 34-02; Secondary 34A55, 34L25, 65J22, 70F17

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Table of Contents

Front/Back Matter

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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

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