# 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

### Table of Contents

**Front/Back Matter**

**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