Exponential lower bound for the eigenvalues of the time-frequency localization operator before the plunge region

Abstract

We prove that the eigenvalues ${\lambda }_n(c)$ of the time-frequency localization operator satisfy ${\lambda }_n(c)>1-{\delta }^c$ for $n=[(1-\epsilon )c]$, where $\delta =\delta (\epsilon )<1$ and $\epsilon >0$ is arbitrary, improving on the result of Bonami, Jaming and Karoui, who proved it for $\epsilon \ge 0.42$. The proof is based on the properties of the Bargmann transform.

13 pages