Central limit theorem for spectral partial bergman kernels

Steven Morris Zelditch, Peng Zhou*

*Corresponding author for this work

Research output: Contribution to journalArticle

3 Citations (Scopus)

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

Fingerprint

Bergman Kernel
Central limit theorem
Subspace
Density of States
Partial
Edge Effects
Gaussian Function
Law of large numbers
Orthogonal Projection
Error function
Density Profile
Otto Toeplitz
Line Bundle
Quantization
Scaling
kernel

Keywords

  • Interface asymptotics
  • Partial Bergman kernel
  • Toeplitz operator

ASJC Scopus subject areas

  • Geometry and Topology

Cite this

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title = "Central limit theorem for spectral partial bergman kernels",
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 K{\"a}hler 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.",
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Central limit theorem for spectral partial bergman kernels. / Zelditch, Steven Morris; Zhou, Peng.

In: Geometry and Topology, Vol. 23, No. 4, 01.01.2019, p. 1961-2004.

Research output: Contribution to journalArticle

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

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KW - Toeplitz operator

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