Central limit theorem for spectral partial bergman kernels

Steven Morris Zelditch, Peng Zhou*

*Corresponding author for this work

Research output: Contribution to journalArticle

3 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)1961-2004
Number of pages44
JournalGeometry and Topology
Volume23
Issue number4
DOIs
StatePublished - Jan 1 2019

Keywords

  • Interface asymptotics
  • Partial Bergman kernel
  • Toeplitz operator

ASJC Scopus subject areas

  • Geometry and Topology

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