Sharp uniqueness and stability of solution for an inverse source problem for the Schr\"odinger equation

TL;DR

This paper proves the uniqueness and stability of determining a spatial source term in the Schr"odinger equation using boundary data, employing integral transforms and null controllability techniques without geometric constraints.

Contribution

It introduces a novel approach using integral transforms and null controllability to establish uniqueness and stability for inverse source problems in Schr"odinger equations without geometric restrictions.

Findings

Logarithmic conditional stability of the Cauchy problem

Uniqueness of solutions with data on small boundary parts

No geometric constraints required for the boundary or time interval

Abstract

The manuscript is concerned with uniqueness and stability for inverse source problem of determining spatially varying factor $f(x)$ of a source term given by $R(t)f(x)$ with suitable given $R(t)$ in the right hand side of the Schr\"odinger equation with time independent coefficients. In order to establish these results we provide a simple proof of a logarithmic conditional stability of the Cauchy problem for the Schr\"odinger equation with time-independent coefficients and the zero Dirichlet boundary conditions on the whole boundary and a proof of uniqueness of solution to the Cauchy problem for the Schr\"odinger equation with data on an arbitrary small part of a lateral boundary. We do not assume any geometrical constraints on subboundary and a time interval is arbitrary. The key is an integral transform, with a kernel solving a null controllability problem for the 1-D Schr\"odinger, which changes a solution of the Schr\"odinger equation to a solution of an elliptic equation.