Dynamic inverse problem for complex Jacobi matrices

TL;DR

The paper studies a dynamic inverse problem for a semi-infinite complex Jacobi matrix in discrete time, formulating boundary-control dynamics and the dynamic response operator $R^T$. It extends the Boundary Control method to non-selfadjoint, one-dimensional discrete systems, deriving a Duhamel-like forward representation and introducing the connecting operator $C^T$, which is determined by the inverse data via $C^T_{ij}=\sum_{k=0}^{T-\max(i,j)} r_{|i-j|+2k}$. Two BC-based inverse approaches are developed: Krein equations, yielding a relation $C^T f^T = a_0[\beta\varkappa^T - \alpha (R^T_{\#})^* \varkappa^T]$, and the Factorization method, yielding explicit formulas for $(a_k)^2$ and $b_k$ in terms of determinants of $C$-blocks; in both, only squares of the off-diagonal coefficients are recoverable, and sign information requires extra data. A complete data-characterization is provided: the given dynamic data correspond to a forward system if and only if all blocks $C^{\,T-k}$ are isomorphisms, ensuring consistent reconstruction across levels. These results advance discrete dynamic inverse problems and inform the limitations of coefficient recovery in non-selfadjoint settings.

Abstract

We consider the inverse dynamic problem for a dynamical system with discrete time associated with a semi-infinite complex Jacobi matrix. We propose two approaches of recovering coefficients from dynamic response operator and answer a question on the characterization of dynamic inverse data.

Theorems & Definitions (18)

• Lemma 1
• proof
• Definition 1
• Definition 2
• Lemma 2
• proof
• Lemma 3
• Theorem 1
• proof
• Remark 1