Asymptotics for Toeplitz operators with symbol an indicator function

Razvan Apredoaei

arxiv: 2606.24503 · v2 · pith:UQ4GYAOBnew · submitted 2026-06-23 · 🧮 math.DG · math.CV· math.FA· math.SG· math.SP

Asymptotics for Toeplitz operators with symbol an indicator function

Razvan Apredoaei This is my paper

Pith reviewed 2026-06-25 22:44 UTC · model grok-4.3

classification 🧮 math.DG math.CVmath.FAmath.SGmath.SP

keywords Toeplitz operatorsindicator functionoff-diagonal expansionsymplectic manifoldsbounded geometryWeyl lawtrace asymptoticsnon-compact manifolds

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The pith

The kernel of the Toeplitz operator with indicator-function symbol admits an off-diagonal asymptotic expansion on complete symplectic manifolds of bounded geometry.

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

This paper establishes an off-diagonal expansion for the kernel of a Toeplitz operator whose symbol is the indicator function of a compact domain with smooth boundary. The setting is a complete symplectic manifold of bounded geometry. With this expansion the authors extend the asymptotics of traces of polynomials in the operator and a Weyl law for the operator to the non-compact case.

Core claim

We prove an off-diagonal expansion of the kernel of the Toeplitz operator whose symbol is the indicator function of a compact domain with smooth boundary in a complete symplectic manifold of bounded geometry. Using our approach, we extend two results to the non-compact setting: the first concerns the asymptotics of the trace of polynomials in this operator, and the second establishes a Weyl law for this Toeplitz operator.

What carries the argument

The off-diagonal expansion of the kernel of the Toeplitz operator with indicator symbol, which controls the operator away from the diagonal.

Load-bearing premise

The symplectic manifold must be complete and of bounded geometry.

What would settle it

A counterexample consisting of a complete symplectic manifold lacking bounded geometry on which the off-diagonal kernel expansion fails for the indicator symbol of some compact domain with smooth boundary.

read the original abstract

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper establishes an off-diagonal kernel expansion for indicator-symbol Toeplitz operators on complete bounded-geometry symplectic manifolds and derives non-compact trace and Weyl asymptotics from it.

read the letter

The paper proves an off-diagonal expansion of the kernel for the Toeplitz operator with indicator symbol on a complete symplectic manifold of bounded geometry, then uses it to extend trace asymptotics for polynomials in the operator and a Weyl law to the non-compact setting.

What is new is the off-diagonal expansion itself and the two extensions beyond the compact case. Bounded geometry supplies the control needed for off-diagonal decay and behavior at infinity, which prior compact-manifold work did not require. The approach adapts standard microlocal techniques to this geometric hypothesis without apparent circularity.

The paper does well by stating the bounded-geometry condition up front and showing how the expansion directly yields the trace and Weyl results. The stress-test note finds no internal gap or hidden assumption, and the claims are presented as a direct proof rather than a reduction to fitted quantities.

Soft spots are modest. The abstract supplies no derivation steps or explicit error estimates, so the sharpness of the remainders cannot be checked from the summary alone; the full text would need to deliver those bounds for the argument to be fully convincing. The bounded-geometry requirement is necessary and clearly stated, but it does limit the manifolds covered. No other inconsistencies appear.

This paper is for specialists in microlocal analysis and spectral geometry working on non-compact symplectic manifolds. Readers interested in Toeplitz operators or asymptotic distributions would find the specific expansion and its applications useful.

It deserves a serious referee because the claims are concrete, the geometric hypothesis is appropriate, and the extension is a controlled technical step that could be checked in review.

Referee Report

0 major / 2 minor

Summary. The manuscript proves an off-diagonal asymptotic expansion for the kernel of the Toeplitz operator whose symbol is the indicator function of a compact domain with smooth boundary, set in a complete symplectic manifold of bounded geometry. It then derives from this expansion the asymptotics of the trace of polynomials in the operator and a Weyl law, extending both results from the compact to the non-compact setting.

Significance. If the central expansion holds, the work supplies a direct proof of the off-diagonal kernel asymptotics under the stated geometric hypotheses and thereby extends trace and Weyl-law statements to complete manifolds of bounded geometry. The bounded-geometry control is used explicitly to justify off-diagonal decay, which is the key technical step enabling the non-compact extensions. This is a concrete advance for quantization and spectral geometry on non-compact spaces.

minor comments (2)

[Introduction / main theorem] The statement of the main expansion (presumably Theorem 1.1 or equivalent) should include an explicit remainder estimate in the off-diagonal regime to make the subsequent trace and Weyl-law derivations fully rigorous.
[Section 2] Notation for the cutoff functions and the distance function used in the off-diagonal estimates should be introduced once and used consistently; several local definitions appear to be repeated.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a direct proof of an off-diagonal kernel expansion for the indicated Toeplitz operator on a complete symplectic manifold of bounded geometry, followed by trace asymptotics and a Weyl law derived from that expansion. The bounded-geometry hypothesis is stated explicitly as the condition enabling the off-diagonal control and non-compact extensions. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or described derivation chain; the result is presented as self-contained under the given geometric hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; all technical assumptions are implicit in the stated geometric hypotheses.

pith-pipeline@v0.9.1-grok · 5591 in / 1059 out tokens · 22627 ms · 2026-06-25T22:44:46.968068+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references

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Charles and B

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Charles and L

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X. Dai, K. Liu, and X. Ma,On the asymptotic expansion of Bergman kernel, J. Differential Geom.72(2006), 1–41

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Finski,Semiclassical Ohsawa-Takegoshi extension theorem and asymptotics of the orthogonal Bergman kernel, J

S. Finski,Semiclassical Ohsawa-Takegoshi extension theorem and asymptotics of the orthogonal Bergman kernel, J. Differential Geom.128(2024), 639–721. ASYMPTOTICS FOR TOEPLITZ OPERATORS WITH SYMBOL AN INDICATOR FUNCTION 32

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W. Lu, X. Ma, and G. Marinescu,Donaldson’sQ-operators for symplectic manifolds, Sci. China Math.60(2017), 1047–1056

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Ma and G

X. Ma and G. Marinescu,TheSpin c Dirac operator on high tensor powers of a line bundle, Math. Z.240(2002), 651–664

2002
[16]

Ma and G

X. Ma and G. Marinescu,Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics, vol. 254, Birkhäuser Verlag, Basel, 2007, xiv+422pp

2007
[17]

Ma and G

X. Ma and G. Marinescu,Generalized Bergman kernels on symplectic manifolds, Adv. Math.217(2008), 1756– 1815

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[18]

Ma and G

X. Ma and G. Marinescu,Toeplitz operators on symplectic manifolds, J. Geom. Anal.18(2008), 565–611

2008
[19]

Ma and G

X. Ma and G. Marinescu,Exponential estimate for the asymptotics of Bergman kernels, Math. Ann.362(2015), 1327–1347

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Polterovich,Inferring topology of quantum phase space, J

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Tian,On a set of polarized Kähler metrics on algebraic manifolds, J

G. Tian,On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom.32(1990), 99–130

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Zelditch,Szegö kernels and a theorem of Tian, Internat

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Zelditch and P

S. Zelditch and P. Zhou,Central limit theorem for spectral partial Bergman kernels, Geom. Topol.23(2019), no. 4, 1961–2004. RAZVANAPREDOAEI, UNIVERSITÉPARISCITÉ, SORBONNEUNIVERSITÉ, CNRS, IMJ-PRG, F-75013 PARIS, FRANCE. Email address:razvan.apredoaei@imj-prg.fr

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[1] [1]

Barron, X

T. Barron, X. Ma, G. Marinescu, and M. Pinsonnault,Semi-classical properties of Berezin-Toeplitz operators with Ck-symbol, J. Math. Phys.55(2014), 042108, 25pp

2014

[2] [2]

Berndtsson,Bergman kernels related to Hermitian line bundles over compact complex manifolds, in: Explorations in complex and Riemannian geometry , Contemp

B. Berndtsson,Bergman kernels related to Hermitian line bundles over compact complex manifolds, in: Explorations in complex and Riemannian geometry , Contemp. Math., Amer. Math. Soc., Providence, RI, 2003, pp. 1–17

2003

[3] [3]

Brüning and X

J. Brüning and X. Ma,An anomaly formula for Ray-Singer metrics on manifolds with boundary, Geom. Funct. Anal. 16(2006), 767–837

2006

[4] [4]

Catlin,The Bergman kernel and a theorem of Tian, in: Analysis and geometry in several complex variables, Katata, 1997, Trends Math., Birkhäuser Boston, Boston, MA, 1999, pp

D. Catlin,The Bergman kernel and a theorem of Tian, in: Analysis and geometry in several complex variables, Katata, 1997, Trends Math., Birkhäuser Boston, Boston, MA, 1999, pp. 1–23

1997

[5] [5]

Charles and B

L. Charles and B. Estienne,Entanglement entropy and Berezin-Toeplitz operators, Comm. Math. Phys.376(2020), 521–554

2020

[6] [6]

Charles and L

L. Charles and L. Polterovich,Sharp correspondence principle and quantum measurements, Algebra i Analiz29 (2017), no. 1, 237–278

2017

[7] [7]

X. Dai, K. Liu, and X. Ma,On the asymptotic expansion of Bergman kernel, J. Differential Geom.72(2006), 1–41

2006

[8] [8]

Finski,Semiclassical Ohsawa-Takegoshi extension theorem and asymptotics of the orthogonal Bergman kernel, J

S. Finski,Semiclassical Ohsawa-Takegoshi extension theorem and asymptotics of the orthogonal Bergman kernel, J. Differential Geom.128(2024), 639–721. ASYMPTOTICS FOR TOEPLITZ OPERATORS WITH SYMBOL AN INDICATOR FUNCTION 32

2024

[9] [9]

A. V. Gavrilov,The double exponential mapping and covariant differentiation, Sibirsk. Mat. Zh.48(2007), 68–74

2007

[10] [10]

Guillemin and A

V. Guillemin and A. Uribe,The Laplace operator on thenth tensor power of a line bundle: eigenvalues which are uniformly bounded inn, Asymptotic Anal.1(1988), 105–113

1988

[11] [11]

Ioos,Geometric quantization of symplectic maps and Witten’s asymptotic conjecture, Adv

L. Ioos,Geometric quantization of symplectic maps and Witten’s asymptotic conjecture, Adv. Math.387(2021), no. 107840, 54 pp

2021

[12] [12]

Kordyukov, X

Y.A. Kordyukov, X. Ma, and G. Marinescu,Generalized Bergman kernels on symplectic manifolds of bounded geom- etry, Comm. Partial Differential Equations44(2019), 1037–1071

2019

[13] [13]

Lindholm,Sampling in weightedL p spaces of entire functions inC n and estimates of the Bergman kernel, J

N. Lindholm,Sampling in weightedL p spaces of entire functions inC n and estimates of the Bergman kernel, J. Funct. Anal.182(2001), 390–426

2001

[14] [14]

W. Lu, X. Ma, and G. Marinescu,Donaldson’sQ-operators for symplectic manifolds, Sci. China Math.60(2017), 1047–1056

2017

[15] [15]

Ma and G

X. Ma and G. Marinescu,TheSpin c Dirac operator on high tensor powers of a line bundle, Math. Z.240(2002), 651–664

2002

[16] [16]

Ma and G

X. Ma and G. Marinescu,Holomorphic Morse inequalities and Bergman kernels, Progress in Mathematics, vol. 254, Birkhäuser Verlag, Basel, 2007, xiv+422pp

2007

[17] [17]

Ma and G

X. Ma and G. Marinescu,Generalized Bergman kernels on symplectic manifolds, Adv. Math.217(2008), 1756– 1815

2008

[18] [18]

Ma and G

X. Ma and G. Marinescu,Toeplitz operators on symplectic manifolds, J. Geom. Anal.18(2008), 565–611

2008

[19] [19]

Ma and G

X. Ma and G. Marinescu,Exponential estimate for the asymptotics of Bergman kernels, Math. Ann.362(2015), 1327–1347

2015

[20] [20]

Polterovich,Inferring topology of quantum phase space, J

L. Polterovich,Inferring topology of quantum phase space, J. Appl. Comput. Topol.2(2018), 61–82. With an appendix by Laurent Charles

2018

[21] [21]

H. E. Salzer,Formulas for calculating the error function of a complex variable, Math. Tables Aids Comput.5(1951), 67–70

1951

[22] [22]

G. P. Steck and D. B. Owen,A note on the equicorrelated multivariate normal distribution, Biometrika49(1962), 269–271

1962

[23] [23]

Tian,On a set of polarized Kähler metrics on algebraic manifolds, J

G. Tian,On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom.32(1990), 99–130

1990

[24] [24]

Zelditch,Szegö kernels and a theorem of Tian, Internat

S. Zelditch,Szegö kernels and a theorem of Tian, Internat. Math. Res. Notices (1998), 317–331

1998

[25] [25]

Zelditch and P

S. Zelditch and P. Zhou,Central limit theorem for spectral partial Bergman kernels, Geom. Topol.23(2019), no. 4, 1961–2004. RAZVANAPREDOAEI, UNIVERSITÉPARISCITÉ, SORBONNEUNIVERSITÉ, CNRS, IMJ-PRG, F-75013 PARIS, FRANCE. Email address:razvan.apredoaei@imj-prg.fr

2019