Pointwise Weyl Law for Partial Bergman Kernels

  • Steve ZelditchEmail author
  • Peng Zhou
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 269)


This article is a continuation of a series by the authors on partial Bergman kernels and their asymptotic expasions. We prove a 2-term pointwise Weyl law for semi-classical spectral projections onto sums of eigenspaces of spectral width \(\hbar =k^{-1}\) of Toeplitz quantizations \(\hat{H}_k\) of Hamiltonians on powers \(L^k\) of a positive Hermitian holomorphic line bundle \(L \rightarrow M\) over a Kähler manifold. The first result is a complete asymptotic expansion for smoothed spectral projections in terms of periodic orbit data. When the orbit is ‘strongly hyperbolic’ the leading coefficient defines a uniformly continuous measure on \({\mathbb R}\) and a semi-classical Tauberian theorem implies the 2-term expansion. As in previous works in the series, we use scaling asymptotics of the Boutet-de-Monvel–Sjostrand parametrix and Taylor expansions to reduce the proof to the Bargmann–Fock case.


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Authors and Affiliations

  1. 1.Department of MathematicsNorthwestern UniversityEvanstonUSA

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