TY - JOUR

T1 - Central limit theorem for spectral partial bergman kernels

AU - Zelditch, Steve

AU - Zhou, Peng

N1 - Funding Information:
Acknowledgements Research partially supported by NSF grants DMS-1541126 and DMS-1810747 and by the Stefan Bergman trust.
Publisher Copyright:
© 2019, Mathematical Sciences Publishers. All rights reserved.

PY - 2019

Y1 - 2019

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

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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 -