### Abstract

Partial Bergman kernels Π_{k,E} are kernels of orthogonal projections onto subspaces S_{k} ⊂ H^{0} (M, L_{k}) of holomorphic sections of the k^{th} power of an ample line bundle over a Kähler manifold (M, ω). The subspaces of this article are spectral subspaces {Ĥ_{k} ≤ E} of the Toeplitz quantization Ĥ_{k} of a smooth Hamiltonian H:M→R. It is shown that the relative partial density of states satisfies Π_{k,E}(z)/Πk(z)→1A where A={H < E}. Moreover it is shown that this partial density of states exhibits “Erf” asymptotics along the interface ∂A; that is, the density profile asymptotically has a Gaussian error function shape interpolating between the values 1 and 0 of 1A. Such “Erf” asymptotics are a universal edge effect. The different types of scaling asymptotics are reminiscent of the law of large numbers and the central limit theorem.

Original language | English (US) |
---|---|

Pages (from-to) | 1961-2004 |

Number of pages | 44 |

Journal | Geometry and Topology |

Volume | 23 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2019 |

### Fingerprint

### Keywords

- Interface asymptotics
- Partial Bergman kernel
- Toeplitz operator

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Geometry and Topology*,

*23*(4), 1961-2004. https://doi.org/10.2140/gt.2019.23.1961

}

*Geometry and Topology*, vol. 23, no. 4, pp. 1961-2004. https://doi.org/10.2140/gt.2019.23.1961

**Central limit theorem for spectral partial bergman kernels.** / Zelditch, Steven Morris; Zhou, Peng.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Central limit theorem for spectral partial bergman kernels

AU - Zelditch, Steven Morris

AU - Zhou, Peng

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Partial Bergman kernels Πk,E are kernels of orthogonal projections onto subspaces Sk ⊂ H0 (M, Lk) of holomorphic sections of the kth power of an ample line bundle over a Kähler manifold (M, ω). The subspaces of this article are spectral subspaces {Ĥk ≤ E} of the Toeplitz quantization Ĥk of a smooth Hamiltonian H:M→R. It is shown that the relative partial density of states satisfies Πk,E(z)/Πk(z)→1A where A={H < E}. Moreover it is shown that this partial density of states exhibits “Erf” asymptotics along the interface ∂A; that is, the density profile asymptotically has a Gaussian error function shape interpolating between the values 1 and 0 of 1A. Such “Erf” asymptotics are a universal edge effect. The different types of scaling asymptotics are reminiscent of the law of large numbers and the central limit theorem.

AB - Partial Bergman kernels Πk,E are kernels of orthogonal projections onto subspaces Sk ⊂ H0 (M, Lk) of holomorphic sections of the kth power of an ample line bundle over a Kähler manifold (M, ω). The subspaces of this article are spectral subspaces {Ĥk ≤ E} of the Toeplitz quantization Ĥk of a smooth Hamiltonian H:M→R. It is shown that the relative partial density of states satisfies Πk,E(z)/Πk(z)→1A where A={H < E}. Moreover it is shown that this partial density of states exhibits “Erf” asymptotics along the interface ∂A; that is, the density profile asymptotically has a Gaussian error function shape interpolating between the values 1 and 0 of 1A. Such “Erf” asymptotics are a universal edge effect. The different types of scaling asymptotics are reminiscent of the law of large numbers and the central limit theorem.

KW - Interface asymptotics

KW - Partial Bergman kernel

KW - Toeplitz operator

UR - http://www.scopus.com/inward/record.url?scp=85069741906&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85069741906&partnerID=8YFLogxK

U2 - 10.2140/gt.2019.23.1961

DO - 10.2140/gt.2019.23.1961

M3 - Article

AN - SCOPUS:85069741906

VL - 23

SP - 1961

EP - 2004

JO - Geometry and Topology

JF - Geometry and Topology

SN - 1465-3060

IS - 4

ER -