Pointwise weyl law for partial bergman kernels

Steven Morris Zelditch*, Peng Zhou

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contribution

1 Citation (Scopus)

Abstract

This article is a continuation of a series by the authors on partial Bergman kernels and their asymptotic expasions. We prove a 2-term pointwise Weyl law for semi-classical spectral projections onto sums of eigenspaces of spectral width ħ = k -1 of Toeplitz quantizations (Formula presented) of Hamiltonians on powers L k of a positive Hermitian holomorphic line bundle L → M over a Kähler manifold. The first result is a complete asymptotic expansion for smoothed spectral projections in terms of periodic orbit data. When the orbit is ‘strongly hyperbolic’ the leading coefficient defines a uniformly continuous measure on R and a semi-classical Tauberian theorem implies the 2-term expansion. As in previous works in the series, we use scaling asymptotics of the Boutet-de-Monvel–Sjostrand parametrix and Taylor expansions to reduce the proof to the Bargmann–Fock case.

Original languageEnglish (US)
Title of host publicationAlgebraic and Analytic Microlocal Analysis - AAMA, Evanston, Illinois, USA, 2012 and 2013
EditorsMichael Hitrik, Dmitry Tamarkin, Boris Tsygan, Steve Zelditch
PublisherSpringer New York LLC
Pages589-634
Number of pages46
ISBN (Print)9783030015862
DOIs
StatePublished - Jan 1 2018
EventWorkshop on Algebraic and Analytic Microlocal Analysis, AAMA 2013 - Evanston, United States
Duration: May 20 2013May 24 2013

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume269
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Other

OtherWorkshop on Algebraic and Analytic Microlocal Analysis, AAMA 2013
CountryUnited States
CityEvanston
Period5/20/135/24/13

Fingerprint

Spectral Projection
Bergman Kernel
Partial
Tauberian theorem
Uniformly continuous
Series
Eigenspace
Levenberg-Marquardt
Taylor Expansion
Otto Toeplitz
Line Bundle
Term
Periodic Orbits
Continuation
Asymptotic Expansion
Quantization
Orbit
Scaling
Imply
Coefficient

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Zelditch, S. M., & Zhou, P. (2018). Pointwise weyl law for partial bergman kernels. In M. Hitrik, D. Tamarkin, B. Tsygan, & S. Zelditch (Eds.), Algebraic and Analytic Microlocal Analysis - AAMA, Evanston, Illinois, USA, 2012 and 2013 (pp. 589-634). (Springer Proceedings in Mathematics and Statistics; Vol. 269). Springer New York LLC. https://doi.org/10.1007/978-3-030-01588-6_13
Zelditch, Steven Morris ; Zhou, Peng. / Pointwise weyl law for partial bergman kernels. Algebraic and Analytic Microlocal Analysis - AAMA, Evanston, Illinois, USA, 2012 and 2013. editor / Michael Hitrik ; Dmitry Tamarkin ; Boris Tsygan ; Steve Zelditch. Springer New York LLC, 2018. pp. 589-634 (Springer Proceedings in Mathematics and Statistics).
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Zelditch, SM & Zhou, P 2018, Pointwise weyl law for partial bergman kernels. in M Hitrik, D Tamarkin, B Tsygan & S Zelditch (eds), Algebraic and Analytic Microlocal Analysis - AAMA, Evanston, Illinois, USA, 2012 and 2013. Springer Proceedings in Mathematics and Statistics, vol. 269, Springer New York LLC, pp. 589-634, Workshop on Algebraic and Analytic Microlocal Analysis, AAMA 2013, Evanston, United States, 5/20/13. https://doi.org/10.1007/978-3-030-01588-6_13

Pointwise weyl law for partial bergman kernels. / Zelditch, Steven Morris; Zhou, Peng.

Algebraic and Analytic Microlocal Analysis - AAMA, Evanston, Illinois, USA, 2012 and 2013. ed. / Michael Hitrik; Dmitry Tamarkin; Boris Tsygan; Steve Zelditch. Springer New York LLC, 2018. p. 589-634 (Springer Proceedings in Mathematics and Statistics; Vol. 269).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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AB - This article is a continuation of a series by the authors on partial Bergman kernels and their asymptotic expasions. We prove a 2-term pointwise Weyl law for semi-classical spectral projections onto sums of eigenspaces of spectral width ħ = k -1 of Toeplitz quantizations (Formula presented) of Hamiltonians on powers L k of a positive Hermitian holomorphic line bundle L → M over a Kähler manifold. The first result is a complete asymptotic expansion for smoothed spectral projections in terms of periodic orbit data. When the orbit is ‘strongly hyperbolic’ the leading coefficient defines a uniformly continuous measure on R and a semi-classical Tauberian theorem implies the 2-term expansion. As in previous works in the series, we use scaling asymptotics of the Boutet-de-Monvel–Sjostrand parametrix and Taylor expansions to reduce the proof to the Bargmann–Fock case.

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Zelditch SM, Zhou P. Pointwise weyl law for partial bergman kernels. In Hitrik M, Tamarkin D, Tsygan B, Zelditch S, editors, Algebraic and Analytic Microlocal Analysis - AAMA, Evanston, Illinois, USA, 2012 and 2013. Springer New York LLC. 2018. p. 589-634. (Springer Proceedings in Mathematics and Statistics). https://doi.org/10.1007/978-3-030-01588-6_13