Uncertainty principle, minimal escape velocities and observability inequalities for schr\"{o}dinger equations

TL;DR

This paper introduces a new method for deriving observability inequalities for Schrödinger equations by combining an uncertainty principle with asymptotic behavior analysis, advancing understanding of controllability and unique continuation.

Contribution

It presents a novel abstract approach that links uncertainty principles with minimal velocity estimates to establish observability inequalities for Schrödinger equations.

Findings

Established a Nazarov type uncertainty principle for Schrödinger operators.

Derived observability inequalities at two time points using asymptotic analysis.

Connected observability results to unique continuation and controllability issues.

Abstract

We develop a new abstract derivation of the observability inequalities at two points in time for Schr\"odinger type equations. Our approach consists of two steps. In the first step we prove a Nazarov type uncertainty principle associated with a non-negative self-adjoint operator $H$ on $L^2(\mathbb{R}^n)$. In the second step we use results on asymptotic behavior of $e^{-itH}$, in particular, minimal velocity estimates introduced by Sigal and Soffer. Such observability inequalities are closely related to unique continuation problems as well as controllability for the Schr\"odinger equation.