Growth of Sobolev norms and controllability of Schr\"odinger equation

TL;DR

This paper proves that under certain conditions, the Schrödinger equation's solutions can grow unbounded in Sobolev norms when influenced by a random potential, highlighting implications for controllability and stabilization.

Contribution

It establishes stabilization results for Schrödinger equations with generic potentials and demonstrates unbounded Sobolev norm growth under random time-dependent potentials.

Findings

Stabilization achieved under generic potential assumptions

Solutions become almost surely unbounded in Sobolev spaces with random potentials

Results have implications for controllability of quantum systems

Abstract

In this paper we obtain a stabilization result for the Schr\"odinger equation under generic assumptions on the potential. Then we consider the Schr\"odinger equation with a potential which has a random time-dependent amplitude. We show that if the distribution of the amplitude is sufficiently non-degenerate, then any trajectory of system is almost surely non-bounded in Sobolev spaces.