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Spectral asymptotics of semi-classical Toeplitz operators on Levi non-degenerate CR manifolds

Wei-Chuan Shen

TL;DR

This work develops a comprehensive semi-classical spectral theory for Toeplitz operators on Levi-nondegenerate CR manifolds by combining Szegő projection microlocality with complex-phase Fourier integral operator calculus. It proves that for a formal self-adjoint first-order pseudodifferential $P$, the semi-classical cutoff $\chi(k^{-1}T_{P,\lambda}^{(q)})$ admits a two-phase off-diagonal expansion when $q=n_-$, with phases $\varphi_-(x,y)$ and $\varphi_+(x,y)$ and explicit leading amplitudes tied to the Levi form via $|\det\mathcal{L}_x|$ and the volume densities; for $q\notin\{n_-,n_+\}$ the projector vanishes. The analysis hinges on a resolvent-parametrix construction and Helffer–Sjöstrand functional calculus, yielding $O(k^{-N})$ remainders and detailed local pictures including the leading diagonal terms $A_0^\pm$. The paper also specializes to circle actions, deriving Bergman-kernel-type expansions for globally free and locally free actions, and shows convergence of scaled spectral measures to a natural continuous limit. Together, these results extend semi-classical Berezin–Toeplitz quantization to CR geometry, with applications to quantization on circle bundles and Sasakian/orbifold settings.

Abstract

We consider any compact CR manifold whose Levi form is non-degenerate of constant signature $(n_-,n_+)$, $n_-+n_+=n$. For $λ>0$ and $q\in\{0,\cdots,n\}$, we let $Π_λ^{(q)}$ be the spectral projection of the Kohn Laplacian of $(0,q)$-forms corresponding to the interval $[0,λ]$. For certain classical pseudodifferential operators $P$, we study a class of generalized elliptic Toeplitz operators $T_{P,λ}^{(q)}:=Π_λ^{(q)}\circ P\circ Π_λ^{(q)}$. For any cut-off $χ\in\mathscr C^\infty_c(\mathbb R\setminus\{0\})$, we establish the full asymptotics of the semi-classical spectral projector $χ(k^{-1}T_{P,λ}^{(q)})$ as $k\to+\infty$. Our main result conclude that the smooth Schwartz kernel $χ(k^{-1}T_{P,λ}^{(n_-)})(x,y)$ is the sum of two semi-classical oscillatory integrals with complex-valued phase functions.

Spectral asymptotics of semi-classical Toeplitz operators on Levi non-degenerate CR manifolds

TL;DR

This work develops a comprehensive semi-classical spectral theory for Toeplitz operators on Levi-nondegenerate CR manifolds by combining Szegő projection microlocality with complex-phase Fourier integral operator calculus. It proves that for a formal self-adjoint first-order pseudodifferential , the semi-classical cutoff admits a two-phase off-diagonal expansion when , with phases and and explicit leading amplitudes tied to the Levi form via and the volume densities; for the projector vanishes. The analysis hinges on a resolvent-parametrix construction and Helffer–Sjöstrand functional calculus, yielding remainders and detailed local pictures including the leading diagonal terms . The paper also specializes to circle actions, deriving Bergman-kernel-type expansions for globally free and locally free actions, and shows convergence of scaled spectral measures to a natural continuous limit. Together, these results extend semi-classical Berezin–Toeplitz quantization to CR geometry, with applications to quantization on circle bundles and Sasakian/orbifold settings.

Abstract

We consider any compact CR manifold whose Levi form is non-degenerate of constant signature , . For and , we let be the spectral projection of the Kohn Laplacian of -forms corresponding to the interval . For certain classical pseudodifferential operators , we study a class of generalized elliptic Toeplitz operators . For any cut-off , we establish the full asymptotics of the semi-classical spectral projector as . Our main result conclude that the smooth Schwartz kernel is the sum of two semi-classical oscillatory integrals with complex-valued phase functions.
Paper Structure (13 sections, 30 theorems, 249 equations)

This paper contains 13 sections, 30 theorems, 249 equations.

Table of Contents

  1. Preliminaries
  2. Introduction and main results
  3. Elements of microlocal and semi-classical analysis
  4. Non-degenerate Cauchy--Riemann manifolds
  5. Szegő projections for lower energy forms
  6. Toeplitz operators for lower energy forms
  7. Fourier integral operators of Szegő type
  8. Microlocal analysis of Toeplitz operators
  9. Expansion of resolvent type Toeplitz operators
  10. Semi-classical asymptotic expansion for the spectral operator
  11. Helffer--Sjöstrand formula and the semi-classical estimates
  12. An example with the globally free circle action
  13. An example with the locally free circle action

Key Result

Theorem 1.1

We let $(X,T^{1,0}X)$ be a compact, non-degenerate CR manifold, and $\dim_\mathbb{R} X:=2n+1$ for $n\geq 1$. We let $\bm\alpha$ be the contact form on $X$ such that the Levi form has the constant signature $(n_-,n_+)$. For $q\in\{0,\cdots,n\}$ and any formally self-adjoint $P\in L^1_{\rm cl}(X;T^{*0 and for each coordinate patch $(\Omega,x)$ in $X$ we have the off-diagonal asymptotic expansion of

Theorems & Definitions (54)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Definition 2.1
  • ...and 44 more