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The Folland-Stein spectrum of some Heisenberg Bieberbach manifolds

Yoshiaki Suzuki

TL;DR

This work extends the spectral analysis of the Folland--Stein operator $\\mathscr{L}_{\alpha}$ to 3D Heisenberg Bieberbach manifolds, by adapting Folland's Weil--Brezin transform framework to nonlattice quotients. For the two nonlattice examples $\\Gamma_{2l,\pi}$ and $\\Gamma'_{2l,\frac{\pi}{2}}$, eigenfunctions on the quotient are obtained as invariants of corresponding lattice eigenfunctions, yielding explicit eigenvalue sets consisting of a harmonic-oscillator family $\frac{\\pi|n|}{2}(2\\lambda+1-\\alpha\\mathrm{sgn}n)$ with multiplicities $l|n|$, and lattice-torus eigenvalues $\\pi^2(\mu^2+\\nu^2)$. The authors prove Weyl-type laws for these quotients, showing the eigenvalue counting grows like $A_{\alpha}\\mathrm{vol}(\\mathcal{M})t^2$, with volume factors reflecting the covers (halved for $\\Gamma_{2l,\pi}$ and quartered for $\\Gamma'_{2l,\frac{\pi}{2}}$). The results illuminate how infra-nilmanifold structure and unitary automorphisms influence the Folland--Stein spectrum in CR geometry.

Abstract

We study the eigenvalues and eigenfunctions of the Folland-Stein operator $\mathscr{L}_α$ on some examples of 3-dimensional Heisenberg Bieberbach manifolds, that is, compact quotients $Γ\backslash\mathbb{H}$ of the Heisenberg group $\mathbb{H}$ by a discrete torsion-free subgroup $Γ$ of $\mathbb{H}\rtimes U(1)$.

The Folland-Stein spectrum of some Heisenberg Bieberbach manifolds

TL;DR

This work extends the spectral analysis of the Folland--Stein operator to 3D Heisenberg Bieberbach manifolds, by adapting Folland's Weil--Brezin transform framework to nonlattice quotients. For the two nonlattice examples and , eigenfunctions on the quotient are obtained as invariants of corresponding lattice eigenfunctions, yielding explicit eigenvalue sets consisting of a harmonic-oscillator family with multiplicities , and lattice-torus eigenvalues . The authors prove Weyl-type laws for these quotients, showing the eigenvalue counting grows like , with volume factors reflecting the covers (halved for and quartered for ). The results illuminate how infra-nilmanifold structure and unitary automorphisms influence the Folland--Stein spectrum in CR geometry.

Abstract

We study the eigenvalues and eigenfunctions of the Folland-Stein operator on some examples of 3-dimensional Heisenberg Bieberbach manifolds, that is, compact quotients of the Heisenberg group by a discrete torsion-free subgroup of .
Paper Structure (7 sections, 14 theorems, 137 equations)

This paper contains 7 sections, 14 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Heisenberg manifolds
  3. The Heisenberg group and lattice subgroups
  4. Harmonic analysis on Heisenberg manifolds
  5. Heisenberg Bieberbach groups
  6. Eigenvalues and eigenfunctions of $\mathscr{L}_\alpha$ on $\Gamma_{2l,\pi}\backslash\mathbb{H}$
  7. Eigenvalues and eigenfunctions of $\mathscr{L}_\alpha$ on $\Gamma'_{2l,\frac{\pi}{2}} \backslash\mathbb{H}$

Key Result

Proposition 2.2

For any lattice $N\subset\mathbb{H}$, there exist a unique $l\in\mathbb{Z}_{>0}$ and an automorphism $\Phi\in\mathop{\mathrm{Aut}}\nolimits_0(\mathbb{H})$ such that $\Phi(N)=N_l$.

Theorems & Definitions (25)

  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3: Brezin brezin
  • Corollary 2.4
  • Proposition 2.5: Folland folland
  • Definition 3.1
  • Remark 3.2
  • Example 3.3
  • Proposition 4.1
  • proof
  • ...and 15 more