Inverse Problems in the Theory of Small Oscillations: (Translations of Mathematical Monographs)

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.

Table of Contents:
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

About the Author :
Vladimir Marchenko, National Academy of Sciences of Ukraine, Kharkiv, Ukraine. Victor Slavin, National Academy of Sciences of Ukraine, Kharkiv, Ukraine.