### 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) |
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Pages (from-to) | 1961-2004 |

Number of pages | 44 |

Journal | Geometry and Topology |

Volume | 23 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2019 |

### Keywords

- Interface asymptotics
- Partial Bergman kernel
- Toeplitz operator

### ASJC Scopus subject areas

- Geometry and Topology

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## Cite this

*Geometry and Topology*,

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