Pith. sign in

REVIEW 3 major objections 4 minor 61 references

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

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read On any compact Levi non-degenerate CR manifold, the spectral function of a Levi-elliptic Toeplitz operator has a complete asymptotic expansion into two oscillatory integrals with complex phases.

desk verdict Real extension of the semiclassical Toeplitz program to Levi non-degenerate CR manifolds; needs one clarification on the quoted Szegő structure theorem's hypotheses in the exceptional signature case. read the letter →

arxiv 2505.13014 v1 pith:7HOPRCCQ submitted 2025-05-19 math.CV math.APmath.SP

Wei-Chuan Shen This is my paper · ORCID
classification math.CVmath.APmath.SP MSC 32V0532A2535P2058J40
keywords CRmanifoldsLevinon-degenerateToeplitzoperatorssemiclassicalspectralasymptoticsSzegőprojectionKohnLaplacianFourierintegrallowerenergyforms
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes that, on any compact CR manifold whose Levi form is non-degenerate with constant signature $(n_-,n_+)$, the spectral calculus of a generalized Toeplitz operator built from a formally self-adjoint first-order pseudodifferential operator has a complete semiclassical asymptotic expansion. At the critical form degree $q=n_-$, and under the Levi-ellipticity conditions (1.1.3)--(1.1.4), the Schwartz kernel of $\chi(k^{-1}T_{P,\lambda}^{(q)})$ is, up to $O(k^{-\infty})$, the sum of two semiclassical oscillatory integrals whose phases are the two complex Szegő phases $\varphi_-$ and $\varphi_+$. The leading diagonal coefficient is computed explicitly from the Levi determinant, the contact-form geometry, and the principal symbol of $P$. For forms of other degrees the same operator is smoothing and hence asymptotically invisible. If the central claim is right, this gives the CR counterpart of Bergman kernel expansions for high powers in mixed-curvature complex geometry, together with a Szegő-type Weyl law for these operators.

What carries the argument

The load-bearing object is the microlocal decomposition of the spectral projection onto lower-energy $(0,q)$-forms (quoted Theorem 1.5): at $q=n_-$, $\Pi_\lambda^{(q)}\equiv S_-+S_+$ modulo smoothing, where $S_\pm$ are Fourier integral operators with complex phases $\varphi_\pm$ satisfying $\operatorname{Im}\varphi_\pm\ge0$, $\varphi_\pm(x,x)=0$, and $d_x\varphi_\pm(x,x)=-d_y\varphi_\pm(x,x)=\mp\alpha(x)$, with $S_+=0$ when $n_-\neq n_+$. On this base the paper builds a parametrix for $z-T_{P,\lambda}^{(q)}$ inside the same complex-phase calculus and feeds it into the almost-analytic functional calculus for $\chi(k^{-1}T_{P,\lambda}^{(q)})$. Complex stationary phase then converts the resolvent-type Toeplitz kernels into semiclassical oscillatory integrals whose phases are rescaled by the principal-symbol factors $p_{I_0,I_0}(-\alpha)$ and $p_{J_0,J_0}(\alpha)$; the Levi determinant enters through the leading Szegő symbols (1.1.15)--(1.1.16).

What would settle it

Take a compact CR manifold with a transversal CR circle action and a first-order $P$ satisfying (1.1.3)--(1.1.4), compute the left side of (1.1.11) from the explicitly known Fourier components of the Szegő projection, and compare the $O(k^{n+1})$ diagonal coefficient with $\int_0^\infty t^n\chi(t)\,dt\,|\det L_x|/(2\pi^{n+1})\,(v(x)/m(x))$; a mismatch at a single point, or a surviving $k^{n}$ term from the wrong phase, would falsify the leading coefficient of Theorem 1.2.

Watch

Extended reading notes

Core claim

Theorem 1.1 asserts that for $q=n_-$ the kernel $\chi(k^{-1}T_{P,\lambda}^{(q)})(x,y)$ equals, modulo $O(k^{-\infty})$ on each coordinate patch, the sum of $\int_0^\infty e^{ikt\varphi_-(x,y)}A_-(x,y,t;k)\,dt$ and $\int_0^\infty e^{ikt\varphi_+(x,y)}A_+(x,y,t;k)\,dt$, with $A_\mp(x,y,t;k)\sim\sum_{j\ge0}A^\mp_j(x,y,t)k^{n+1-j}$ and $A_+$ identically zero unless $n_-=n_+$. At the diagonal the leading symbol is $A^-_0(x,x,t)=t^n\chi(p_{I_0,I_0}(-\alpha_x)t)\,|\det L_x|/(2\pi^{n+1})\,(v(x)/m(x))\,\tau_x^{n_-}$, with the mirror formula $A^+_0(x,x,t)=t^n\chi(p_{J_0,J_0}(\alpha_x)t)\,|\det L_x|/(2\pi^{n+1})\,(v(x)/m(x))\,\tau_x^{n_+}$ when $n_-=n_+$. This is a full asymptotic expansion of the spectral projector, not merely a counting law: every coefficient $A^\mp_j$ has a well-defined symbol class, and the phases carry the off-diagonal structure.

Load-bearing premise

The whole two-phase expansion rests on the quoted structure theorem that the spectral projection onto lower-energy $(0,n_-)$-forms is, modulo smoothing, exactly the sum of two complex-phase oscillatory integral operators $S_-+S_+$ and is smoothing for other degrees; if that decomposition is false or requires additional hypotheses such as $L^2$-closed range, the main expansion loses its stated two-phase form.

Editorial extensions

If this is right

  • For $q\notin\{n_-,n_+\}$, the operator $\chi(k^{-1}T_{P,\lambda}^{(q)})$ is smoothing for large $k$, so the semiclassical spectral function vanishes to infinite order and the Toeplitz operator is compact.
  • For $q=n_-$, the kernel admits a full off-diagonal expansion in powers $k^{n+1-j}$; away from the diagonal the positive imaginary parts of the phases make the kernel rapidly decreasing in $k$.
  • On the diagonal, the leading coefficient is explicit in local geometry: it is proportional to $|\det L_x|$, the volume ratio $v(x)/m(x)$, the projector $\tau_x^{n_-}$, and the cutoff $\chi$ evaluated at $p_{I_0,I_0}(-\alpha_x)t$; an analogous positive-phase term appears when $n_-=n_+$.
  • The scaled spectral measures $k^{-n-1}\sum_j\delta(t-k^{-1}\lambda_j)$ converge weakly to the absolutely continuous measure $C_P^{(q)}t^n\,dt$ of Corollary 1.4, giving a Szegő-type Weyl law.
  • The spectrum of $T_{P,\lambda}^{(q)}$ at $q=n_-$ is discrete with accumulation only at $\pm\infty$, so the cutoff $\chi$ cleanly separates the positive and negative spectral sides, which the two-phase expansion captures separately.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The two-phase structure suggests that the positive and negative spectral ends are governed by two distinct effective Hamiltonians $p_{I_0,I_0}(-\alpha)$ and $p_{J_0,J_0}(\alpha)$; a natural continuation would be a two-sided quantum ergodicity statement for eigenfunctions on each end, which the paper does not address.
  • The $O(k^{-\infty})$ vanishing outside $q\in\{n_-,n_+\}$ can be read as a spectral refinement of the classical coarse vanishing theorem for high powers of line bundles: in the CR picture, low-energy Toeplitz modes of other form degrees carry no leading semiclassical trace at all.
  • For CR manifolds with transversal circle action, the parametrix should reproduce known orbifold Bergman kernel expansions on each orbit-type stratum; the paper works out the globally free and locally free cases in Section 3, but a systematic treatment of the singular strata as an independent derivation would make the two-phase form directly checkable.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies generalized Toeplitz operators T_{P,λ}^{(q)} = Π_λ^{(q)} P Π_λ^{(q)} on compact Levi non-degenerate CR manifolds, where Π_λ^{(q)} is the spectral projection of the Kohn Laplacian on (0,q)-forms for the interval [0,λ]. For q = n_- and P satisfying the Levi-ellipticity conditions (1.1.3)-(1.1.4), the main theorem claims that for any χ ∈ C_c^∞(R\{0}) the Schwartz kernel of χ(k^{-1}T_{P,λ}^{(q)}) is, modulo O(k^{-∞}), the sum of two semi-classical oscillatory integrals with complex phases φ_- and φ_+, with a complete asymptotic expansion in powers k^{n+1-j} and explicit leading coefficients (Theorems 1.1 and 1.2). A Szegő-type limit theorem for the scaled spectral measure and two examples involving free and locally free circle actions are also given. The proof is built on the quoted structure theorem of Hsiao-Marinescu for lower-energy Szegő projections (Theorem 1.5), followed by a parametrix construction for (z - T_{P,λ}^{(q)}) and a Helffer-Sjöstrand resolvent calculation.

Significance. If the result is correct, it substantially extends semi-classical Toeplitz spectral asymptotics from the strictly pseudoconvex setting to Levi non-degenerate CR manifolds of arbitrary constant signature, with explicit parameter-free leading coefficients and concrete applications to CR circle actions. The paper gives precise theorem statements, a complete proof architecture in the style of Melin-Sjöstrand and Boutet de Monvel, and worked examples in §3.2-§3.3. The main results are, however, conditional on the quoted structure theorem and on a dense sequence of microlocal estimates; the two load-bearing points identified below require explicit verification or repair before the central claim can be regarded as established.

major comments (3)
  1. [§1.4, Theorem 1.1] The main expansion is logically inherited from the quoted structure theorem Theorem 1.5, yet the exceptional signature case is left ambiguous. The text states that the two-FIO description of the true Szegő projection Π^{(q)} is available only when □_b^{(q)} has L^2-closed range, that closed range is automatic only when |n_- - n_+| ≠ 1, and that a three-dimensional counterexample exists in the exceptional case; Theorem 1.1, however, imposes no closed-range or Y(q) hypothesis. Since Theorems 2.5, 2.17 and 3.5 all invoke Theorem 1.5, the authors should either quote the exact hypotheses of [43, Theorem 4.1] and verify explicitly that it applies to Π_λ^{(q)} for every compact non-degenerate CR manifold when |n_- - n_+| = 1, or add the missing hypothesis to Theorem 1.1 and restrict the corollaries accordingly. A single sentence citing the precise statement of [43, Theorem 4.1] may suffice if the theorem is unconditional, but as written the paper's own §1.4 discussion creates a genuine ambiguity that blocks acceptance.
  2. [§2.3, proof of Theorem 2.17, Eq. (2.3.37)] The proof uses the identity p_{J0,J0}(-α) = -p_{J0,J0}(α) > 0 in the change of variables for the C^{+,+} term. This identity is not a consequence of formal self-adjointness and first-order classicality of P. For example, a formally self-adjoint first-order operator with principal symbol -|ξ| satisfies p(α)<0 but p(-α)=p(α)<0, so it meets (1.1.4) without making p(-α) positive. The subsequent definition of Ψ_+ ∈ Ph(p^{-1}_{J0,J0}(-α)α) requires the phase coefficient Λ to be positive, and the leading coefficients (2.3.15)-(2.3.16) depend on the same change of variables. The proof should either derive the parity identity from the assumptions on P and on the CR structure, or reformulate the parametrix using p_{J0,J0}(α) directly, e.g. by the change of variables t = -β p_{J0,J0}(α_w) in (2.3.37), which requires no assumption on p(-α). This is load-bearing because Theorem 2.17 is the core parametrix underlying the spectral expansion.
  3. [§3.1, Theorem 3.5 and Theorem 1.2] The leading coefficient formulas (3.1.35)-(3.1.36) and (1.1.20)-(1.1.21) are stated without an independent derivation: they are presented as consequences of Theorem 2.17 and the quoted leading symbol computations. In particular, the sign and the argument of χ in the A^+_0 term are only consistent with the rest of the proof under the parity identity discussed in the previous comment. The authors should give the short calculation showing how (3.1.36) follows from Theorem 2.17 within their chosen phase normalization, or, if they adopt the alternative change of variables, update the statements of Theorem 1.2 and (3.1.36) accordingly. This is necessary because Theorem 1.2 is one of the two headline results and its explicit form is used in Corollary 1.4.
minor comments (4)
  1. [§1.1] The cross-references to '§4.1', '§4.2' and '§4.3' in the introduction should be to '§3.1', '§3.2' and '§3.3', respectively.
  2. [Abstract and §2.2] There are small typos: 'Our main result conclude' should be 'Our main result concludes', and 'microloal' should be 'microlocal'.
  3. [§1.1, around (1.1.7)] The phrase 'proplery supported H¨ormander symbol' should read 'properly supported Hörmander symbol'.
  4. [§2.3, Theorem 2.17] The condition on r^+_1 in the statement of Theorem 2.17 appears to contain a typo: it should presumably be 'r^+_1 = 0 when n_- ≠ n_+', consistent with (2.3.34), (2.3.38) and all other statements, rather than 'n_- = n_+' as printed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.1 is a new semiclassical consequence of the quoted Szegő-kernel structure theorem, not a restatement of its inputs.

full rationale

The derivation chain is: the quoted structure theorem for Πλ^(q) (Theorem 1.5, [43, Thm 4.1]) supplies only the two-FIO form and symbol leading terms of the Szegő projection; the paper's own work is the resolvent parametrix for T_{P,λ}=Πλ P Πλ (Theorems 2.6, 2.17, 2.18), the Helffer–Sjöstrand functional calculus, and the k-negligibility estimates (Theorems 3.4, 3.7–3.12) that convert the parametrix into χ(k^{-1}T) with the stated k^{n+1-j} asymptotics. No fitted parameter is introduced and no quantity is defined in terms of the claimed spectral asymptotics; the leading coefficients (1.1.20)–(1.1.21) are computed from the symbol of P and the quoted Szegő symbol formulas (1.1.15)–(1.1.16). The self-citations [32,33] are to the semiclassical method and to prior strictly-pseudoconvex calculations, not to the target theorem, and the one formula cited from [32] (Theorem 2.3) is a standard phase-transformation rule independently grounded in Melin–Sjöstrand/Boutet de Monvel–Sjöstrand. The exceptional-case hypothesis concern about L²-closed range when |n_- - n_+|=1 is a correctness/verification issue about the quoted theorem, not a circular reduction; the paper itself flags the counterexample in [7]. Sections 3.2–3.3 further check the expansion in circle-bundle examples against independent Bergman-kernel asymptotics. Therefore there is no circular step.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on four cited results: the microlocal structure of the lower-energy Szego projection, the Szego phase normal form, Melin-Sjostrand complex stationary phase, and Kohn's L2 range and spectral theory. No free parameters are introduced and no new entities are postulated.

assumptions (4)
  • domain assumption Hsiao-Marinescu lower-energy Szego projection structure theorem (Theorem 1.5, from [43, Theorem 4.1]): for q = n_-, Pi^{(q)}_lambda = S_- + S_+ + F with S_+/- complex-phase FIOs and F smoothing; S_+ = 0 when n_- differs from n_+; for q not in {n_-, n_+}, Pi^{(q)}_lambda is smoothing.
    Invoked in Section 1.4 and used throughout; it is the core structural input for the two-phase oscillatory expansion.
  • standard math Szego phase function normal form and equivalence (Hsiao-Marinescu [43, Theorems 3.4, 5.4], Boutet de Monvel-Sjostrand [10], Melin-Sjostrand [53]): there exist phi_+/- with Im phi_+/- >= 0, phi_+/-(x,y) = 0 iff x = y, d_x phi_+/-(x,x) = -d_y phi_+/-(x,x) = +/- alpha(x), and any two such phase…
    Provides the phases used in (1.1.11) and the composition identities in Theorems 2.3 and 2.4, as well as Remark 1.9.
  • standard math Melin-Sjostrand complex stationary phase theorem [53, Theorem 2.3] with its remainder estimates.
    Used in Theorem 2.17, equations (2.3.34) and (2.3.38), and repeatedly in Section 3 to reduce composed oscillatory integrals.
  • domain assumption Kohn's L2 closed range theorem [47] and the spectral theory of self-adjoint extensions [21], guaranteeing Pi^{(q)}_lambda = Pi^{(q)} for suitable lambda and the discrete spectral structure of T^{(q)}_{P,lambda}.
    Underlies Theorems 2.8 through 2.10 and the spectral expansion formula (1.1.5).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Spectral asymptotics of semi-classical Toeplitz operators on Levi non-degenerate CR manifolds." pith.science (2026). https://pith.science/paper/7HOPRCCQ

@misc{pith2026250513014,
  author       = {Pith},
  title        = {Pith review of: Spectral asymptotics of semi-classical Toeplitz operators on Levi non-degenerate CR manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7HOPRCCQ}},
  note         = {Machine review of arXiv:2505.13014}
}
abstract

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

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

61 extracted references · 57 canonical work pages

  1. [32]

    Herrmann, C.-Y

    H. Herrmann, C.-Y. Hsiao, G. Marinescu, and W.-C. Shen, Semi-classical spectral asymptotics of Toeplitz operators on CR manifolds, arXiv:2303.17319 (2023)

    arXiv 2023

  2. [43]

    Hsiao and G

    C.-Y. Hsiao and G. Marinescu, On the singularities of the Szeg˝ o projections on lower energy forms , J. Differential Geom. 107 (2017), no. 1, 83–155. MR3698235

    work page 2017

  3. [1]

    Andreotti and H

    A. Andreotti and H. Grauert, Th´ eor` eme de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193–259. MR150342

    work page 1962

  4. [2]

    M. S. Baouendi, L. P . Rothschild, and F. Tr`eves, CR structures with group action and extendability of CR functions , Invent. Math. 82 (1985), no. 2, 359–396. MR809720

    work page 1985

  5. [3]

    Berman and J

    R. Berman and J. Sj ¨ostrand, Asymptotics for Bergman-Hodge kernels for high powers of complex line bundles , Ann. Fac. Sci. Toulouse Math. (6) 16 (2007), no. 4, 719–771. MR2789717

    work page 2007

  6. [4]

    Bordemann, E

    M. Bordemann, E. Meinrenken, and M. Schlichenmaier, Toeplitz quantization of K¨ ahler manifolds andgl(N), N→ ∞ limits, Comm. Math. Phys. 165 (1994), no. 2

    work page 1994

  7. [5]

    Borthwick, T

    D. Borthwick, T. Paul, and A. Uribe, Semiclassical spectral estimates for Toeplitz operators , Ann. Inst. Fourier (Grenoble) 48 (1998), no. 4, 1189–1229. MR1656013

    work page 1998

  8. [6]

    Boutet de Monvel, Hypoelliptic operators with double characteristics and related pseudo-differential operators , Comm

    L. Boutet de Monvel, Hypoelliptic operators with double characteristics and related pseudo-differential operators , Comm. Pure Appl. Math. 27 (1974), 585–639. MR370271

    work page 1974

Show all 61 references
  1. [7]

    L. Boutet de Monvel, Int´ egration des ´ equations de Cauchy-Riemann induites formelles, S´eminaire Goulaouic-Lions- Schwartz 1974–1975: ´Equations aux d´eriv´ees partielles lin´eaires et non lin´eaires, pp. Exp. No. 9, 14. ´Ecole Poly- tech., Centre de Math., Paris, 1975. MR0409893

    1974

  2. [8]

    Boutet de Monvel, On the index of Toeplitz operators of several complex variables , Invent

    L. Boutet de Monvel, On the index of Toeplitz operators of several complex variables , Invent. Math. 50 (1978/79), no. 3, 249–272. MR520928

    1978

  3. [9]

    Boutet de Monvel and V

    L. Boutet de Monvel and V . Guillemin, The spectral theory of Toeplitz operators , Annals of Mathematics Studies, vol. 99, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1981. MR620794

    1981

  4. [10]

    Boutet de Monvel and J

    L. Boutet de Monvel and J. Sj ¨ostrand, Sur la singularit´ e des noyaux de Bergman et de Szeg˝ o, Journ ´ees ´Equations aux D ´eriv´ees Partielles de Rennes (1975), pp. 123–164. Ast ´erisque, No. 34–35, Soc. Math. France, Paris, 1976. MR590106

    1975

  5. [11]

    C. P . Boyer and K. Galicki, Sasakian geometry , Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008. MR2382957

    2008

  6. [12]

    Catlin, The Bergman kernel and a theorem of Tian, Analysis and geometry in several complex variables (Katata, 1997), pp

    D. Catlin, The Bergman kernel and a theorem of Tian, Analysis and geometry in several complex variables (Katata, 1997), pp. 1–23. Birkh¨auser Boston, Boston, MA, 1999. MR1699887

    1997

  7. [13]

    Chang, H

    C.-C. Chang, H. Herrmann, and H. Chin-Yu, On the second coefficient in the semi-classical expansion of Toeplitz Operators on CR manifolds, arXiv:2412.11697 (2024)

    2024 arXiv

  8. [14]

    Chang and A

    R. Chang and A. Rabinowitz, Scaling asymptotics for Szeg˝ o kernels on Grauert tubes , J. Geom. Anal. 33 (2023), no. 2, Paper No. 60, 27. MR4523282

    2023

  9. [15]

    Charles, Berezin-Toeplitz operators, a semi-classical approach , Comm

    L. Charles, Berezin-Toeplitz operators, a semi-classical approach , Comm. Math. Phys. 239 (2003), no. 1-2, 1–28. MR1997113

    2003

  10. [16]

    Charles, On the spectrum of nondegenerate magnetic Laplacians , Anal

    L. Charles, On the spectrum of nondegenerate magnetic Laplacians , Anal. PDE 17 (2024), no. 6, 1907–1952. MR4776289

    2024

  11. [17]

    Chen and M.-C

    S.-C. Chen and M.-C. Shaw, Partial differential equations in several complex variables, AMS/IP Studies in Advanced Mathematics, vol. 19, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001. MR1800297

    2001

  12. [18]

    T. C. Collins and G. Sz ´ekelyhidi, K-semistability for irregular Sasakian manifolds, J. Differential Geom. 109 (2018), no. 1, 81–109. MR3798716

    2018

  13. [19]

    X. Dai, K. Liu, and X. Ma, On the asymptotic expansion of Bergman kernel , J. Differential Geom. 72 (2006), no. 1, 1–41. MR2215454

    2006

  14. [20]

    X. Dai, K. Liu, and X. Ma, A remark on weighted Bergman kernels on orbifolds , Math. Res. Lett. 19 (2012), no. 1, 143–148. MR2923181

    2012

  15. [21]

    E. B. Davies, Spectral theory and differential operators , Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge University Press, Cambridge, 1995. MR1349825

    1995

  16. [22]

    Dimassi and J

    M. Dimassi and J. Sj ¨ostrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, vol. 268, Cambridge University Press, Cambridge, 1999. MR1735654

    1999

  17. [23]

    Drewitz, B

    A. Drewitz, B. Liu, and G. Marinescu, Toeplitz operators and zeros of square-integrable random holomorphic sections, arXiv 2404.15983 (2024)

    2024 arXiv

  18. [24]

    J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982), no. 2, 259–268. MR674406

    1982

  19. [25]

    Finski, Small eigenvalues of Toeplitz operators, Lebesgue envelopes and Mabuchi geometry, arXiv:2502.01554 (2025)

    S. Finski, Small eigenvalues of Toeplitz operators, Lebesgue envelopes and Mabuchi geometry, arXiv:2502.01554 (2025)

    2025 arXiv

  20. [26]

    Galasso and C.-Y

    A. Galasso and C.-Y. Hsiao, Toeplitz operators on CR manifolds and group actions, J. Geom. Anal. 33 (2023), no. 1, Paper No. 21, 55. MR4510165

    2023

  21. [27]

    Galasso and C.-Y

    A. Galasso and C.-Y. Hsiao, Functional calculus and quantization commutes with reduction for Toeplitz operators on CR manifolds, Math. Z. 308 (2024), no. 1, Paper No. 5. MR4779004

    2024

  22. [28]

    Grauert, ¨Uber Modifikationen und exzeptionelle analytische Mengen, Math

    H. Grauert, ¨Uber Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331–368. MR137127

    1962

  23. [29]

    Grigis and J

    A. Grigis and J. Sj ¨ostrand, Microlocal analysis for differential operators: An introduction , London Mathematical Society Lecture Note Series, vol. 196, Cambridge University Press, Cambridge, 1994. MR1269107

    1994

  24. [30]

    Herrmann, C.-Y

    H. Herrmann, C.-Y. Hsiao, and X. Li, Szeg¨ o kernel asymptotic expansion on strongly pseudoconvex CR manifolds with S1 action, Internat. J. Math. 29 (2018), no. 9, 1850061, 35. MR3845397 SPECTRAL ASYMPTOTICS OF SEMI-CLASSICAL LEVI ELLIPTIC TOEPLITZ OPERATORS 40

    2018

  25. [31]

    Herrmann, C.-Y

    H. Herrmann, C.-Y. Hsiao, and X. Li, Torus equivariant Szeg˝ o kernel asymptotics on strongly pseudoconvex CR manifolds, Acta Math. Vietnam. 45 (2020), no. 1, 113–135. MR4081369

    2020

  26. [33]

    Herrmann, C.-Y

    H. Herrmann, C.-Y. Hsiao, G. Marinescu, and W.-C. Shen, Induced Fubini-Study metrics on strictly pseudoconvex CR manifolds and zeros of random CR functions, arXiv:2401.09143 (2024)

    2024 arXiv

  27. [34]

    H ¨ormander, Fourier integral operators

    L. H ¨ormander, Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79–183. MR388463

    1971

  28. [35]

    H ¨ormander, The analysis of linear partial differential operators

    L. H ¨ormander, The analysis of linear partial differential operators. I , Classics in Mathematics, Springer-Verlag, Berlin, 2003. Distribution theory and Fourier analysis, Reprint of the second (1990) edition. MR1996773

    1990

  29. [36]

    H ¨ormander, The null space of the ∂-Neumann operator, Ann

    L. H ¨ormander, The null space of the ∂-Neumann operator, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 5, 1305– 1369, xiv, xx. MR2127850

    2004

  30. [37]

    Hsiao, Projections in several complex variables, M´em

    C.-Y. Hsiao, Projections in several complex variables, M´em. Soc. Math. Fr. (N.S.) 123 (2010), 131. MR2780123

    2010

  31. [38]

    Hsiao, On the coefficients of the asymptotic expansion of the kernel of Berezin-Toeplitz quantization, Ann

    C.-Y. Hsiao, On the coefficients of the asymptotic expansion of the kernel of Berezin-Toeplitz quantization, Ann. Global Anal. Geom. 42 (2012), no. 2, 207–245. MR2947953

    2012

  32. [39]

    Hsiao, The second coefficient of the asymptotic expansion of the weighted Bergman kernel for (0, q) forms on Cn, Bull

    C.-Y. Hsiao, The second coefficient of the asymptotic expansion of the weighted Bergman kernel for (0, q) forms on Cn, Bull. Inst. Math. Acad. Sin. (N.S.) 11 (2016), no. 3, 521–570. MR3585390

    2016

  33. [40]

    Hsiao and R.-T

    C.-Y. Hsiao and R.-T. Huang, G-invariant Szeg˝ o kernel asymptotics and CR reduction, Calc. Var. Partial Differential Equations 60 (2021), no. 1, Paper No. 47, 48. MR4210746

    2021

  34. [41]

    Hsiao and G

    C.-Y. Hsiao and G. Marinescu, Asymptotics of spectral function of lower energy forms and Bergman kernel of semi- positive and big line bundles, Comm. Anal. Geom. 22 (2014), no. 1, 1–108. MR3194375

    2014

  35. [42]

    Hsiao and G

    C.-Y. Hsiao and G. Marinescu, Berezin-Toeplitz quantization for lower energy forms , Comm. Partial Differential Equations 42 (2017), no. 6, 895–942. MR3683308

    2017

  36. [44]

    Hsiao and G

    C.-Y. Hsiao and G. Marinescu, Semi-classical spectral asymptotics of Toeplitz operators on strictly pseudodonvex domains, The Bergman kernel and related topics, [2024] ©2024, pp. 239–259. Springer Proc. Math. Stat., 447, Springer, Singapore. MR4731755

    2024

  37. [45]

    L. Ioos, V . Kaminker, L. Polterovich, and D. Shmoish,Spectral aspects of the Berezin transform, Ann. H. Lebesgue 3 (2020), 1343–1387. MR4191393

    2020

  38. [46]

    A. V . Karabegov and M. Schlichenmaier, Identification of Berezin-Toeplitz deformation quantization , J. Reine Angew. Math. 540 (2001), 49–76. MR1868597

    2001

  39. [47]

    J. J. Kohn, The range of the tangential Cauchy-Riemann operator, Duke Math. J. 53 (1986), no. 2, 525–545. MR850548

    1986

  40. [48]

    Loi and G

    A. Loi and G. Placini, Any Sasakian structure is approximated by embeddings into spheres , Forum Mathematicum (Aug. 2024)

    2024

  41. [49]

    Lu, The second coefficient of the asymptotic expansion of the Bergman kernel of the Hodge-Dolbeault operator , J

    W. Lu, The second coefficient of the asymptotic expansion of the Bergman kernel of the Hodge-Dolbeault operator , J. Geom. Anal. 25 (2015), no. 1, 25–63. MR3299268

    2015

  42. [50]

    Ma and G

    X. Ma and G. Marinescu, The first coefficients of the asymptotic expansion of the Bergman kernel of the Spinc Dirac operator, Internat. J. Math. 17 (2006), no. 6, 737–759. MR2246888

    2006

  43. [51]

    Ma and G

    X. Ma and G. Marinescu, Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics, vol. 254, Birkh¨auser Verlag, Basel, 2007. MR2339952

    2007

  44. [52]

    Ma and G

    X. Ma and G. Marinescu, Berezin-Toeplitz quantization on K¨ ahler manifolds, J. Reine Angew. Math. 662 (2012), 1–56. MR2876259

    2012

  45. [53]

    Melin and J

    A. Melin and J. Sj ¨ostrand, Fourier integral operators with complex-valued phase functions, Fourier integral operators and partial differential equations (Colloq. Internat., Univ. Nice, Nice, 1974), pp. 120–223. Lecture Notes in Math., Vol. 459, Springer-Verlag, Berlin-New Yo...

    1974

  46. [54]

    Menikoff and J

    A. Menikoff and J. Sj ¨ostrand, On the eigenvalues of a class of hypoelliptic operators , Math. Ann. 235 (1978), no. 1, 55–85. MR481627

    1978

  47. [55]

    Paoletti, On the Weyl law for Toeplitz operators, Asymptot

    R. Paoletti, On the Weyl law for Toeplitz operators, Asymptot. Anal. 63 (2009), no. 1-2, 85–99. MR2524535

    2009

  48. [56]

    Ross and R

    J. Ross and R. Thomas, Weighted Bergman kernels on orbifolds , J. Differential Geom. 88 (2011), no. 1, 87–107. MR3019812

    2011

  49. [57]

    Ross and R

    J. Ross and R. Thomas, Weighted projective embeddings, stability of orbifolds, and constant scalar curvature K¨ ahler metrics, J. Differential Geom. 88 (2011), no. 1, 109–159. MR2819757

    2011

  50. [58]

    Shabtai, Off-diagonal estimates of partial Bergman kernels on S 1-symmetric K¨ ahler manifolds, arXiv 2401.15416 (2024)

    O. Shabtai, Off-diagonal estimates of partial Bergman kernels on S 1-symmetric K¨ ahler manifolds, arXiv 2401.15416 (2024)

    2024

  51. [59]

    Shiffman and S

    B. Shiffman and S. Zelditch, Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds, J. Reine Angew. Math. 544 (2002), 181–222. MR1887895

    2002

  52. [60]

    Zelditch, Szeg˝ o kernels and a theorem of Tian, Internat

    S. Zelditch, Szeg˝ o kernels and a theorem of Tian, Internat. Math. Res. Notices 6 (1998), 317–331. MR1616718

    1998

  53. [61]

    Zelditch and P

    S. Zelditch and P . Zhou, Central limit theorem for spectral partial Bergman kernels , Geom. Topol. 23 (2019), no. 4, 1961–2004. MR3981005 CY CERGY PARIS UNIVERSITY , SITE SAINT -M ARTIN , 2 AVENUE ADOLPHE CHAUVIN , 95300 P ONTOISE , FRANCE . Email address: wei-chuan.shen@u-ce...

    2019

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.