Sharp estimates for eigenvalues of localization operators with applications to area laws

TL;DR

This paper derives sharp bounds for the eigenvalues of localization operators with applications to area laws, providing new estimates that improve understanding of eigenvalue distributions in Fourier analysis.

Contribution

It introduces new sharp bounds for eigenvalues of localization operators, especially when sets are unions of parallelepipeds, and applies these to trace estimates and area laws.

Findings

Established sharp uniform upper bounds for eigenvalue counting functions.

Provided bounds close to the conjectural optimal in general cases.

Applied bounds to trace estimates, confirming an enhanced area law.

Abstract

We study the eigenvalues of the localization operator $S_{A, B} = P_A\mathcal{F}^{-1}P_B\mathcal{F} P_A$, where $\mathcal{F}$ is the Fourier transform and $A = cA_0, B = B_0$ for some fixed sets $A_0, B_0\subset \mathbb{R}^d$ and a large parameter $c > 0$. For the counting function of the eigenvalues $|\{n: \varepsilon < \lambda_n(A,B)\le 1-\varepsilon\}|$ we obtain a sharp uniform upper bound if one of the sets is a finite disjoint union of parallelepipeds and a bound which is only a single logarithm off the conjectural optimal bound in the general case. These bounds are applied to the estimation of traces ${\rm{Tr}}\, f(S_{A,B})$ for functions $f$ with a very low regularity, in particular establishing an enhanced area law in the former case.