### 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 language | English (US) |
---|---|

Title of host publication | Algebraic and Analytic Microlocal Analysis - AAMA, Evanston, Illinois, USA, 2012 and 2013 |

Editors | Michael Hitrik, Dmitry Tamarkin, Boris Tsygan, Steve Zelditch |

Publisher | Springer New York LLC |

Pages | 589-634 |

Number of pages | 46 |

ISBN (Print) | 9783030015862 |

DOIs | |

State | Published - Jan 1 2018 |

Event | Workshop on Algebraic and Analytic Microlocal Analysis, AAMA 2013 - Evanston, United States Duration: May 20 2013 → May 24 2013 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
---|---|

Volume | 269 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Other

Other | Workshop on Algebraic and Analytic Microlocal Analysis, AAMA 2013 |
---|---|

Country | United States |

City | Evanston |

Period | 5/20/13 → 5/24/13 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*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

}

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Pointwise weyl law for partial bergman kernels

AU - Zelditch, Steven Morris

AU - Zhou, Peng

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=85059853857&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85059853857&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-01588-6_13

DO - 10.1007/978-3-030-01588-6_13

M3 - Conference contribution

SN - 9783030015862

T3 - Springer Proceedings in Mathematics and Statistics

SP - 589

EP - 634

BT - Algebraic and Analytic Microlocal Analysis - AAMA, Evanston, Illinois, USA, 2012 and 2013

A2 - Hitrik, Michael

A2 - Tamarkin, Dmitry

A2 - Tsygan, Boris

A2 - Zelditch, Steve

PB - Springer New York LLC

ER -