Dynamical inverse problem for the discrete Schrödinger operator
A. S. Mikhaylov, A. S. Mikhaylov
TL;DR
This work develops a dynamical inverse problem for the discrete Schrödinger operator on a semi-infinite lattice with discrete time, using the Boundary Control method and the response operator as the inverse data. It derives three complementary reconstruction frameworks—Krein equations, a factorization method, and Gelfand-Levitan equations—along with a rigorous characterization test for inverse data, enabling partial recovery of the potential from boundary measurements. The authors also connect dynamic inverse data to spectral data by presenting a spectral representation of the connecting operator $C^T$ and the response kernel $r_t$, showing how to compute the potential from spectral information via the BC-method. Together, these results establish a discrete-analog of the continuous BC approach and pave the way for inverse problems on Jacobi matrices and semi-infinite systems, including potential extensions to inverse spectral problems.
Abstract
We consider the inverse problem for the dynamical system with discrete Schrödinger operator and discrete time. As an inverse data we take a \emph{response operator}, the natural analog of the dynamical Dirichlet-to-Neumann map. We derive two types of equations of inverse problem and answer a question on the characterization of the inverse data, i.e. we describe the set of operators, which are \emph{response operators} of the dynamical system governed by the discrete Schrödinger operator.