Inverse Problems in the Theory of Small Oscillations
American Mathematical Society (Verlag)
978-1-4704-4890-5 (ISBN)
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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 Schrodinger 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 Schrodinger operators, where only the class of tridiagonal Hermitian matrices are considered.
Vladimir Marchenko, National Academy of Sciences of Ukraine, Kharkiv, Ukraine. Victor Slavin, National Academy of Sciences of Ukraine, Kharkiv, Ukraine.
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
The properties of completely $/textbf{M}$-extendable sets
The 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
The examples of solving numerically the inverse problem of the theory of small oscillations
Bibliography
| Erscheinungsdatum | 15.02.2019 |
|---|---|
| Reihe/Serie | Translations of Mathematical Monographs |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Gewicht | 448 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Analysis |
| ISBN-10 | 1-4704-4890-4 / 1470448904 |
| ISBN-13 | 978-1-4704-4890-5 / 9781470448905 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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