Critical time for the observability of Kolmogorov-type equations

TL;DR

This paper investigates the observability of two-dimensional Kolmogorov-type equations with quadratic degeneracy, establishing bounds for the critical time using Carleman estimates and spectral analysis, with sharp results in symmetric cases.

Contribution

It provides new bounds for the critical observability time of Kolmogorov equations, including sharp results in symmetric scenarios, through advanced spectral and Carleman estimate techniques.

Findings

Derived lower and upper bounds for the critical observability time.

Established sharp bounds in symmetric cases.

Analyzed spectral properties of non-selfadjoint Schrödinger operators.

Abstract

This paper is devoted to the observability of a class of two-dimensional Kolmogorov-type equations presenting a quadratic degeneracy. We give lower and upper bounds for the critical time. These bounds coincide in symmetric settings, giving a sharp result in these cases. The proof is based on Carleman estimates and on the spectral properties of a family of non-selfadjoint Schr\"odinger operators, in particular the localization of the first eigenvalue and Agmon type estimates for the corresponding eigenfunctions.