複流形及軌形上的拓譜立茲量化之半經典漸近

Abstract

在本論文中，我們引入了具有局部性光譜間隙的複流形並利用縮放方法研究其伯格曼核和譜核且得到其漸近。 透過這些漸近，我們給出了滿足局部譜間隙條件的全純厄米軌線叢的伯格曼核的漸近行為。並且，我們透過對縮放伯格曼核的觀察及靜相公式建立了一個伯格曼核及拓譜立茲量化算子全漸近展開。 此外，我們建立了有關擬微分算子的拓譜立茲算子的變形量化。

In this thesis, we introduce complex manifolds with local spectral gap and study their asymptotic behavior by using scaling method. With these asymptotic, we obtain an asymptotic expansion for the Bergman kernel of a Hermitian holomorphic orbifold line bundle satisfying the local spectral gap condition. Furthermore, we establish the full asymptotic expansion of both the Bergman kernel and the Toeplitz operator, using the observations of the scaled Bergman kernel and the stationary phase formula. In addition, we establish the deformation quantization for Toeplitz operators with pseudodifferential operators.