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A note on the local Weyl formula on compact Lie groups

Duván Cardona, Julio Delgado, Michael Ruzhansky

Abstract

In this note we reformulate the spectral side of the Weyl law in the language of the matrix-valued quantisation on compact Lie groups.

A note on the local Weyl formula on compact Lie groups

Abstract

In this note we reformulate the spectral side of the Weyl law in the language of the matrix-valued quantisation on compact Lie groups.
Paper Structure (10 sections, 5 theorems, 47 equations)

This paper contains 10 sections, 5 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Motivation: the quantum ergodicity problem
  3. Main results
  4. Preliminaries
  5. The Fourier analysis of a compact Lie group
  6. The quantisation formula
  7. Notations
  8. The spectral side of the local Weyl formula on $G$
  9. Local Weyl formula for zero order operators
  10. Local Weyl formula for operators of positive order

Key Result

Theorem 1.1

Let $G$ be a compact Lie group of dimension $n,$ and let $A\in \Psi^0(G)$ be a classical pseudo-differential operator of order zero. In terms of the following data: for any $\lambda>0,$ the partial trace of $A$ admits the asymptotic expansion Moreover, the convergence at infinity of its average with respect to the eigenvalue counting function $N(\lambda):=\#\{[\xi]\in \widehat{G}:|\xi|\leq \lambd

Theorems & Definitions (23)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1: Unitary representation of a compact Lie group
  • Remark 2.2: Irreducible representations
  • Definition 2.3: Equivalent representations
  • Definition 2.4: The unitary dual
  • Definition 2.5: Group Fourier transform
  • Remark 2.6: The Fourier inversion formula on a compact Lie group
  • ...and 13 more