Comments on observability and stabilization of magnetic Schr{\"o}dinger equations

TL;DR

This paper extends observability and stabilization results from the Schr{"o}dinger equation to the magnetic case, introducing new stabilization insights using classical methods and elliptic Carleman inequalities.

Contribution

It develops new observability and stabilization results for magnetic Schr{"o}dinger equations, including logarithmic stabilization, using classical multiplier and Carleman techniques.

Findings

Establishes observability inequalities for magnetic Schr{"o}dinger equations.

Proves exponential stabilization under geometric conditions.

Demonstrates logarithmic stabilization results with resolvent estimates.

Abstract

We are mainly interested in extending the known results on ob-servability inequalities and stabilization for the Schr{\"o}dinger equation to the magnetic Schr{\"o}dinger equation. That is in presence of a magnetic potential. We establish observability inequalities and exponential stabilization by extending the usual multiplier method, under the same geometric condition to that needed for the Schr{\"o}dinger equation. We also prove, with the help of elliptic Carleman inequalities, logarithmic stabilization results through a resolvent estimate. Although the approach is classical, these results on logarithmic stabilization seem to be new even for the Schr{\"o}dinger equation.