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Inverse dynamic problems for canonical systems and de Branges spaces

A. S. Mikhaylov, V. S. Mikhaylov

TL;DR

The paper develops a unified framework linking inverse problems for wave, Dirac, and Jacobi-dynamics to inverse problems for canonical systems with a Hamiltonian $H$, by constructing de Branges spaces from dynamic data via the Boundary Control method. It shows that, under suitable smoothness and positivity assumptions, dynamic IPs for these systems are equivalent to IPs for canonical systems governed by $iH\frac{d}{dt}-J\frac{d}{dx}=0$, and it establishes steps to recover $H$ from dynamic data. A key contribution is the dynamic construction of de Branges spaces, including extensions to Dirac-type dynamics to restore controllability and to encode spectral information, with an outline for connecting operators and Fourier transforms to spectral measures. The work posits a dynamic-de Branges correspondence for general Hamiltonians and discusses obstacles (smoothness, rank changes) and potential resolutions via Krein-string analogies, aiming to unify inverse problems across continuous and discrete, as well as Dirac-type and wave-type systems. This framework has potential implications for tomography and inverse spectral problems by providing a common, operator-theoretic approach to recover system parameters from boundary data.

Abstract

We show the equivalence of inverse problems for different dynamical systems and corresponding canonical systems. For canonical system with general Hamiltonian we outline the strategy of studying the dynamic inverse problem and procedure of construction of corresponding de Branges space.

Inverse dynamic problems for canonical systems and de Branges spaces

TL;DR

The paper develops a unified framework linking inverse problems for wave, Dirac, and Jacobi-dynamics to inverse problems for canonical systems with a Hamiltonian , by constructing de Branges spaces from dynamic data via the Boundary Control method. It shows that, under suitable smoothness and positivity assumptions, dynamic IPs for these systems are equivalent to IPs for canonical systems governed by , and it establishes steps to recover from dynamic data. A key contribution is the dynamic construction of de Branges spaces, including extensions to Dirac-type dynamics to restore controllability and to encode spectral information, with an outline for connecting operators and Fourier transforms to spectral measures. The work posits a dynamic-de Branges correspondence for general Hamiltonians and discusses obstacles (smoothness, rank changes) and potential resolutions via Krein-string analogies, aiming to unify inverse problems across continuous and discrete, as well as Dirac-type and wave-type systems. This framework has potential implications for tomography and inverse spectral problems by providing a common, operator-theoretic approach to recover system parameters from boundary data.

Abstract

We show the equivalence of inverse problems for different dynamical systems and corresponding canonical systems. For canonical system with general Hamiltonian we outline the strategy of studying the dynamic inverse problem and procedure of construction of corresponding de Branges space.

Paper Structure

This paper contains 11 sections, 3 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. de Branges spaces.
  3. Dynamical canonical systems for wave equation, Dirac system and Jacobi system with discrete time.
  4. Wave equation with a potential on a half-line.
  5. Wave equation on a half-line.
  6. Dirac system on a half-line.
  7. Semi-infinite Jacobi matrices.
  8. Canonical systems with smooth strictly positive Hamiltonian.
  9. One-velocity wave system.
  10. Dirac-type dynamical system.
  11. Dynamic approach to de Branges spaces.

Key Result

Proposition 1

The solution to (DiracSyst) admits the following representation: where $\tau(s)=\int_0^s\sqrt{d_1(\alpha)d_2(\alpha)}\,d\alpha$ is eikonal, $x(t)$ is a function inverse to $\tau(x)$, the kernel $w=$ is twice differentiable in $\left\{(x,s)\,|\, 0\leqslant\tau(x)\leqslant s\leqslant T\right\}$, $A=$, where $a^{1,2}$ are solutions to the following system

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2
  • Proposition 3