Smoothing effect for the regularized Schr\"odinger equation with non controlled orbits

TL;DR

This paper demonstrates that the smoothing effect and uniform stabilization of the strongly dissipative Schr"odinger equation can be achieved without the geometric control condition, challenging previous assumptions.

Contribution

It shows that the geometric control condition is not required for smoothing and stabilization in the dissipative Schr"odinger equation, providing new insights into control theory.

Findings

Smoothing effect achieved without geometric control condition

Uniform stabilization possible without controlled orbits

Challenges previous assumptions in control theory

Abstract

We prove that the geometric control condition is not necessary to obtain the smoothing effect and the uniform stabilization for the strongly dissipative Schr\"odinger equation.