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Semiclassical analysis on compact nilmanifolds

Veronique Fischer

TL;DR

The paper develops a semiclassical framework for compact nil-manifolds $M=\Gamma\backslash G$, using a group-Fourier based quantization on $M\times \widehat{G}$ and operator-valued symbols to study quantum limits of hypoelliptic operators. It proves the existence of semiclassical measures $\Gamma d\gamma$ describing limits of bounded $L^2(M)$-families as the semiclassical parameter $\varepsilon\to0$, and decomposes these measures according to the infinite- and finite-dimensional parts of $\widehat{G}$. Focusing on eigenfunctions of positive Rockland operators (including sub-Laplacians), it establishes localisation and invariance properties of the quantum limits and identifies obstructions in noncommutative Heisenberg-type nil-manifolds, while torus-like cases behave more classically. The work combines Weyl-type asymptotics, a detailed symbolic calculus, and operator-valued measure techniques to provide a robust framework for quantum limits on nil-manifolds, with potential extensions to Rockland operators perturbed by lower-order terms.

Abstract

In this paper, we define and study semi-classical analysis and semi-classical limits on compact nil-manifolds. As an application, we obtain properties of quantum limits for sub-Laplacians in this context, and more generally for positive Rockland operators.

Semiclassical analysis on compact nilmanifolds

TL;DR

The paper develops a semiclassical framework for compact nil-manifolds , using a group-Fourier based quantization on and operator-valued symbols to study quantum limits of hypoelliptic operators. It proves the existence of semiclassical measures describing limits of bounded -families as the semiclassical parameter , and decomposes these measures according to the infinite- and finite-dimensional parts of . Focusing on eigenfunctions of positive Rockland operators (including sub-Laplacians), it establishes localisation and invariance properties of the quantum limits and identifies obstructions in noncommutative Heisenberg-type nil-manifolds, while torus-like cases behave more classically. The work combines Weyl-type asymptotics, a detailed symbolic calculus, and operator-valued measure techniques to provide a robust framework for quantum limits on nil-manifolds, with potential extensions to Rockland operators perturbed by lower-order terms.

Abstract

In this paper, we define and study semi-classical analysis and semi-classical limits on compact nil-manifolds. As an application, we obtain properties of quantum limits for sub-Laplacians in this context, and more generally for positive Rockland operators.

Paper Structure

This paper contains 55 sections, 33 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on nilpotent Lie groups and compact nil-manifold
  3. About nilpotent Lie groups
  4. Representations of $G$ and $L^1(G)$
  5. The Plancherel formula
  6. The von Neumann algebra and $C^*$-algebra of $G$
  7. Compact nil-manifolds
  8. $\Gamma$-periodic functions on $G$ and functions on $M$
  9. Operators on $M$ and $G$
  10. Preliminaries on Rockland operators on $G$ and $M$
  11. Graded nilpotent Lie group
  12. Definition
  13. Dilations and homogeneity
  14. Homogeneous quasi-norms
  15. Approximation of the identity on $G$ and $M$
  16. ...and 40 more sections

Key Result

Theorem 1.1

We consider $\Gamma$ a discrete co-compact subgroup of a graded nilpotent Lie group $G$, and denote by $M:=\Gamma \backslash G$ the corresponding compact nil-manifold. Let $(\phi_\varepsilon)_{\varepsilon\in (0,1]}$ be a bounded family in $L^2(M)$. There exists a sequence $(\varepsilon_k)_{k\in \mat Moreover, we may assume that the limit $\lim_{k\to \infty} \|\phi_{\varepsilon_k}\|_{L^2(M)}$ exist

Theorems & Definitions (70)

  • Theorem 1.1
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • Remark 3.2
  • Theorem 3.3: Hulanicki's theorem
  • Theorem 3.4
  • Remark 3.5
  • ...and 60 more