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Unified approach to classical equations of inverse problem theory

M. I. Belishev, V. S. Mikhaylov

TL;DR

This work presents a unified boundary control (BC) framework that derives the classical inverse problem integral equations—Gelfand-Levitan, Krein, and Marchenko—via boundary-control problems for wave-type equations. By formulating forward and inverse problems for Dirichlet and Neumann controls on the half-line, the authors show that inverse data, notably the extended boundary data $ ilde{R}^{2T}$ and the response operator $R$, uniquely determine potentials and densities (e.g., $q$ and $ ho$) through Fredholm, Gelfand-Levitan, and Krein-type equations, with locality properties that reflect finite propagation speed. The BC-method establishes explicit connections between control/response operators and the inverse problem data, enabling both reconstruction and visualization of waves, and it provides a natural path to multidimensional generalizations and alternative approaches. Overall, the paper integrates inverse spectral, scattering, and dynamical problems into a single, data-driven, operator-theoretic framework with clear solvability and locality advantages.

Abstract

The boundary control (BC-) method is an approach to inverse problems based upon their deep relations to control and system theory. We show that the classical integral equations of inverse problem theory (Gelfand-Levitan, Krein and Marchenko equations) can be derived in the framework of the BC-method in a unified way. Namely, to solve each of these equations is in fact to solve a relevant boundary control problem, whereas its solution is determined by the inverse data.

Unified approach to classical equations of inverse problem theory

TL;DR

This work presents a unified boundary control (BC) framework that derives the classical inverse problem integral equations—Gelfand-Levitan, Krein, and Marchenko—via boundary-control problems for wave-type equations. By formulating forward and inverse problems for Dirichlet and Neumann controls on the half-line, the authors show that inverse data, notably the extended boundary data and the response operator , uniquely determine potentials and densities (e.g., and ) through Fredholm, Gelfand-Levitan, and Krein-type equations, with locality properties that reflect finite propagation speed. The BC-method establishes explicit connections between control/response operators and the inverse problem data, enabling both reconstruction and visualization of waves, and it provides a natural path to multidimensional generalizations and alternative approaches. Overall, the paper integrates inverse spectral, scattering, and dynamical problems into a single, data-driven, operator-theoretic framework with clear solvability and locality advantages.

Abstract

The boundary control (BC-) method is an approach to inverse problems based upon their deep relations to control and system theory. We show that the classical integral equations of inverse problem theory (Gelfand-Levitan, Krein and Marchenko equations) can be derived in the framework of the BC-method in a unified way. Namely, to solve each of these equations is in fact to solve a relevant boundary control problem, whereas its solution is determined by the inverse data.
Paper Structure (17 sections, 11 theorems, 122 equations)

This paper contains 17 sections, 11 theorems, 122 equations.

Table of Contents

  1. Introduction.
  2. Forward problems
  3. Problem $1_{\rm Dir}$
  4. Problems $1^T_{\rm Dir}$ and $\widetilde{1}^{2T}_{\rm Dir}$
  5. Problems with Neumann control
  6. Boundary control problems
  7. System $\alpha_{\rm Dir}^T$
  8. BCP for $\alpha^T_{\rm Dir}.$
  9. System $\alpha^T_{\rm Neum}.$
  10. BCP for $\alpha^T_{\rm Neum}$.
  11. Inverse problems
  12. IP for $\alpha^T_{\rm Dir}$
  13. IP for $\alpha^T_{\rm Neum}$
  14. Inverse scattering problem
  15. System $\alpha_{\rm scatt}$
  16. ...and 2 more sections

Key Result

Lemma 1

The operator $W^T$ is an isomorphism from $\mathcal{F}^T$ onto $\mathcal{H}^T$. The relation $W^T\mathcal{F}^{T,\,\xi}=\mathcal{H}^\xi$ is valid as $0\leqslant\xi\leqslant T$.

Theorems & Definitions (16)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • ...and 6 more