Inverse dynamic problem for the wave equation with periodic boundary conditions

TL;DR

This work develops a dynamic inverse problem for the wave equation with a periodic potential on $(0,2\pi)$ using a boundary triplet and the dynamic Dirichlet-to-Neumann map as data. By extending the problem to the real line and applying the Boundary Control method, it derives Krein and Gelfand-Levitan type equations that enable reconstruction of the potential from the response data on $[0,2\pi]$, and it clarifies how dynamic data determine spectral objects via the spectral measure and Weyl function. The reconstruction proceeds by first recovering the extended potential $\widetilde{q}$ on $(-\pi,\pi)$ through the Krein or Gelfand-Levitan framework and then extending periodically to obtain $q$ on $(0,2\pi)$; the results bridge dynamic and spectral inverse data for the periodic Schrödinger operator. The findings provide a rigorous link between time-domain boundary measurements and spectral characteristics, with implications for inverse problems on graphs with cycles and nano-scale waveguides, leveraging finite propagation speed and boundary-control techniques. The methodology centers on boundary triplets, connecting operators, and Weyl functions to connect dynamic response with spectral data.

Abstract

We consider the inverse dynamic problem for the wave equation with a potential on an interval $(0,2π)$ with periodic boundary conditions. We use a boundary triplet to set up the initial-boundary value problem. As an inverse data we use a response operator (dynamic Dirichlet-to-Neumann map). Using the auxiliary problem on the whole line, we derive equations of the inverse problem. We also establish the relationships between dynamic and spectral inverse data.

Theorems & Definitions (10)

• Theorem 1
• Remark 1
• Remark 2
• Remark 3
• Corollary 1
• Remark 4
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5