论文标题
复杂投影空间上磁性拉普拉斯的热系数$ \ mathbf {p}^{n}（\ mathbb {C}）
Heat coefficients for magnetic Laplacians on the complex projective space $\mathbf{P}^{n}(\mathbb{C})$
论文作者
论文摘要
该操作员用$Δ_ν$表示与$ν$成比例的均匀磁场强度扰动的fubini-study laplacian，该操作员具有一个离散的频谱，该频谱由eigenvalues $β_m，\ m \ in \ m athbb {z} _+$ in \ mathbb {z} _+$，在限制了$ n $ projective $ n $上。对于相应的特征空间，我们通过直接使用Zaremba的扩展为其再现核提供了新的证明。然后，这些内核用于获得$Δ_ν$的热核的积分表示。使用每个$β_M$的多样性的合适多项式分解，就Jacobi的Theta函数及其高阶衍生物而言，我们写下了与$Δ_ν$相关的热运算符的痕量公式。这样做使我们能够通过在Bernoulli数字和多项式方面提供相应的热系数来确定该迹线的渐近学为$ t \ searrow 0^+$。可以在与$Δ_ν$相关的光谱Zeta函数的分析中利用所获得的结果。
Denoting by $Δ_ν$ the Fubini-Study Laplacian perturbed by a uniform magnetic field strength proportional to $ν$, this operator has a discrete spectrum consisting on eigenvalues $β_m, \ m\in\mathbb{Z}_+$, when acting on bounded functions of the complex projective $n$-space. For the corresponding eigenspaces, we give a new proof for their reproducing kernels by using Zaremba's expansion directly. These kernels are then used to obtain an integral representation for the heat kernel of $Δ_ν$. Using a suitable polynomial decomposition of the multiplicity of each $β_m$, we write down a trace formula for the heat operator associated with $Δ_ν$ in terms of Jacobi's theta functions and their higher order derivatives. Doing so enables us to establish the asymptotics of this trace as $t\searrow 0^+$ by giving the corresponding heat coefficients in terms of Bernoulli numbers and polynomials. The obtained results can be exploited in the analysis of the spectral zeta function associated with $Δ_ν$.
