Dynamical inverse problem for the discrete Schr\"odinger operator

TL;DR

This paper investigates the inverse problem for a discrete Schrödinger operator in a dynamical setting, characterizing the response operators that can arise from such systems and deriving related inverse equations.

Contribution

It introduces a novel characterization of response operators for the discrete Schrödinger inverse problem and derives two types of inverse equations for the dynamical system.

Findings

Characterization of response operators for the discrete Schrödinger inverse problem

Derivation of two types of inverse equations

Identification of conditions for response operators to originate from the dynamical system

Abstract

We consider the inverse problem for the dynamical system with discrete Schr\"odinger 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 Schr\"odinger operator.