Sharp estimates for eigenvalues of localization operators before the plunge region

TL;DR

This paper provides sharp eigenvalue estimates for two types of localization operators, revealing a phase transition in their spectra and highlighting differences in eigenvalue decay near the plunge region.

Contribution

It establishes precise uniform bounds on eigenvalues of localization operators near the critical threshold, demonstrating a qualitative difference between time-frequency and coherent state cases.

Findings

Eigenvalues exhibit a phase transition at the threshold.

Eigenvalues decay exponentially when n < (1-ε)c.

Different asymptotic behaviors for the two localization operators.

Abstract

We study two closely related yet different localization operators: the time-frequency localization operator to the pair of intervals $S_{I, J} = P_I \mathcal{F}^{-1} P_J\mathcal{F} P_I$ and the localization of the coherent state transform to the square $L_Q$. Eigenvalues of both of them exhibit the same phase transition: if $|I| |J| = |Q| = c$ then first $\approx c$ eigenvalues are very close to $1$, then there are $o(c)$ intermediate eigenvalues and the rest of the eigenvalues are very close to $0$. Moreover, for both of them if $n < (1-\varepsilon)c$ for fixed $\varepsilon > 0$ then the eigenvalues are exponentially close to $1$. The goal of this paper is to establish sharp uniform bounds on these eigenvalues when $n$ is close to $c$ and see if there is a qualitative difference between the spectrums of $S_{I, J}$ and $S_Q$. We show that for $n < c -c^{0.99}$, say, in the time-frequency localization case we have $-\log(1-\lambda_n(c))\asymp\frac{c-n}{\log(\frac{2c}{c-n})}$ while in the coherent state transform case we have $-\log(1-\mu_n(c))\asymp (\sqrt{c}-\sqrt{n})^2,$ which is much smaller if $c-n = o(c)$, so there is indeed a difference between these two cases. The proofs crucially rely on the complex-analytic interpretations of these localization operators.