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Inverse problems for finite Jacobi matrices and Krein--Stieltjes strings

Alexander Mikhaylov, Victor Mikhaylov

Abstract

We consider dynamic inverse problems for a dynamical system associated with a finite Jacobi matrix and for a system describing propagation of waves in a finite Krein-Stieltjes string. We offer three methods of recovering unknown parameters: entries of a Jacobi matrix in the first problem and point masses and distances between them in the second, from dynamic Dirichlet-to-Neumann operators. We also answer a question on a characterization of dynamic inverse data for these two problems.

Inverse problems for finite Jacobi matrices and Krein--Stieltjes strings

Abstract

We consider dynamic inverse problems for a dynamical system associated with a finite Jacobi matrix and for a system describing propagation of waves in a finite Krein-Stieltjes string. We offer three methods of recovering unknown parameters: entries of a Jacobi matrix in the first problem and point masses and distances between them in the second, from dynamic Dirichlet-to-Neumann operators. We also answer a question on a characterization of dynamic inverse data for these two problems.
Paper Structure (11 sections, 14 theorems, 127 equations)

This paper contains 11 sections, 14 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. IP for dynamical system associated with finite Jacobi matrix.
  3. Forward problem
  4. Operators of the BC method.
  5. Inverse problem.
  6. Krein equations.
  7. IP for a finite Krein--Stieltjes string.
  8. Forward problem.
  9. Operators of the Boundary control method.
  10. Response operator.
  11. Inverse problem.

Key Result

Lemma 1

The solution to (DnSst) admits the spectral representation where

Theorems & Definitions (27)

  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Proposition 1
  • proof
  • ...and 17 more