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

Aleksei Kulikov, Martin Dam Larsen

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 $|\{n: \varepsilon < λ_n(A,B)\le 1-\varepsilon\}|$ 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.

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

Abstract

We study the eigenvalues of the localization operator , where is the Fourier transform and for some fixed sets and a large parameter . For the counting function of the eigenvalues 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 for functions with a very low regularity, in particular establishing an enhanced area law in the former case.
Paper Structure (23 sections, 27 theorems, 221 equations)

This paper contains 23 sections, 27 theorems, 221 equations.

Table of Contents

  1. Introduction
  2. Time-frequency localization operator
  3. Case of two boxes
  4. General case
  5. Proof strategy
  6. Area laws
  7. Lower order terms for $\operatorname{Tr} S^2$
  8. One-term asymptotics
  9. Previous results on one-dimensional operators
  10. Case of two boxes
  11. General case
  12. Singular values of one-dimensional operators
  13. Singular values of $J_r$
  14. Singular values of $I_r$
  15. Trace class conditions
  16. ...and 8 more sections

Key Result

Theorem 1.1

Consider $A = B = [0, 1]^d$, $d\geq 1$, and $c\geq 2$. There exists $\alpha_d \geq 4$ such that uniformly for all $\alpha_d^{-c}< \varepsilon < 1/2$. Moreover, if $\varepsilon < c^{-\alpha_d}$ then we also have If $\varepsilon \leq \alpha_d^{-c}$ then there are no eigenvalues larger than $1-\varepsilon$ and

Theorems & Definitions (52)

  • Theorem 1.1
  • Proposition 1.2
  • Definition 1
  • Definition 2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Lemma 1.8
  • ...and 42 more