Uniqueness for the Schr\"odinger equation with an inverse square potential and application to controllability and inverse problems

TL;DR

This paper establishes a sharp uniqueness result for the Schrödinger equation with an inverse square potential, enabling advances in controllability and inverse source problems without geometric restrictions.

Contribution

It extends the uniqueness technique to singular Schrödinger equations with inverse square potentials, facilitating new controllability and inverse problem solutions.

Findings

Proves a sharp uniqueness result for singular Schrödinger equations.

Demonstrates approximate controllability using distributed control.

Establishes uniqueness in inverse source problems.

Abstract

In this paper, we prove a sharp uniqueness result for the singular Schr\"odinger equation with an inverse square potential. This will be done without assuming geometrical restrictions on the observation region. The proof relies on a recent technique transforming the Schr\"odinger equation into an elliptic equation. We show that this technique is still applicable for singular equations. In our case, substantial difficulties arise when dealing with singular potentials of cylindrical type. Using the uniqueness result, we show the approximate controllability of the equation using a distributed control. The uniqueness result is also applied to prove the uniqueness for an inverse source problem.