Wednesday, 30 March 2022 15:57

Inverse Problems in the Theory of Small Oscillations

  • Author(s): Vladimir Marchenko, Viktor Slavin (B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences, Kharkiv, Ukraine)
  • Publisher: AMS
  • Year: 2018

[V. Marchenko, V. Slavin, Inverse problems in the theory of small oscillations, AMS, Providence, RI, 2018]

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 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 Schrödinger operators, where only the class of tridiagonal Hermitian matrices are considered.