Unique Continuation Inequalities for the Schr\"odinger equations associated with the Special Hermite operators

TL;DR

This paper establishes unique continuation inequalities for Schr"odinger equations linked to special Hermite operators, showing that smallness at two times outside finite measure sets implies smallness everywhere, with similar results for Hermite-Schr"odinger equations.

Contribution

It introduces new unique continuation inequalities for Schr"odinger equations associated with special Hermite operators, extending understanding of solution behavior.

Findings

Solutions remain small throughout space if small at two times outside finite measure sets.

Results apply to both special Hermite and Hermite-Schr"odinger equations.

Provides a framework for understanding solution propagation and control.

Abstract

We investigate unique continuation inequalities for solutions of the Schr\"odinger equations associated with special Hermite operators. Our main result establishes that if the solution remains small at two distinct time points outside sets of finite measure, then the solution also remains small throughout the entire space. We also explore analogous results for the Hermite-Schr\"odinger equations.