Identification of the Potential Coefficient in the Schrödinger Equation with Incomplete Initial Conditions from a Boundary Observation

This paper deals with an inverse problem of the Schrödinger equation, a fundamental equation in quantum mechanics. Specifically, we focus on incomplete data, where there are missing terms in the potential term and the initial condition. The potential term is a critical part of the equation, representing the potential energy of the system under investigation. Our objective is to obtain valuable information about this potential term without the need to determine the unknown initial condition. To achieve this, we employ the sentinel method, which is a functional that is sensitive to only one unknown and insensitive to others. Our research shows that the existence of this functional is connected to solving an optimal control problem, which we accomplish using the Hilbert Uniqueness Method. By using this approach, we are able to gain insights into the potential coefficient, which can provide significant benefits in a wide range of applications.