Volume 23, issue 4 (2019)

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Central limit theorem for spectral partial Bergman kernels

Steve Zelditch and Peng Zhou

Geometry & Topology 23 (2019) 1961–2004
Abstract

Partial Bergman kernels ${\Pi }_{k,E}$ are kernels of orthogonal projections onto subspaces ${\mathsc{S}}_{k}\subset {H}^{0}\left(M,{L}^{k}\right)$ of holomorphic sections of the power of an ample line bundle over a Kähler manifold $\left(M,\omega \right)$. The subspaces of this article are spectral subspaces $\left\{{Ĥ}_{k}\le E\right\}$ of the Toeplitz quantization ${Ĥ}_{k}$ of a smooth Hamiltonian $H:\phantom{\rule{0.3em}{0ex}}M\to ℝ$. It is shown that the relative partial density of states satisfies ${\Pi }_{k,E}\left(z\right)∕{\Pi }_{k}\left(z\right)\to {1}_{\mathsc{A}}$ where $\mathsc{A}=\left\{H. Moreover it is shown that this partial density of states exhibits “Erf” asymptotics along the interface $\partial \mathsc{A}$; that is, the density profile asymptotically has a Gaussian error function shape interpolating between the values $1$ and $0$ of ${1}_{\mathsc{A}}$. 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.

Keywords
Toeplitz operator, partial Bergman kernel, interface asymptotics
Mathematical Subject Classification 2010
Primary: 32A60, 32L10, 81Q50