复射影空间$\mathbf{P}^{n}(\mathbb{C})$上磁拉普拉斯算子的热系数
Heat coefficients for magnetic Laplacians on the complex projective space $\mathbf{P}^{n}(\mathbb{C})$
摘要：
用$Δ_ν$表示受与$ν$成比例的均匀磁场强度扰动的Fubini研究拉普拉斯算子，当作用于复射影$n$ -空间的有界函数时，该算子具有由本征值$β_m, \ m\in\mathbb{Z}_+$组成的离散谱。对于相应的特征空间，我们直接利用Zaremba展开式给出了再生核的一个新的证明。这些内核，然后用于获得一个积分表示的热核的$Δ_ν$。使用一个合适的多项式分解的每个$β_m$的多重性，我们写下来的跟踪公式与$Δ_ν$的Jacobi的θ函数和它们的高阶导数方面的热运营商。这样做使我们能够通过给出相应的伯努利数和多项式的热系数来建立该迹的渐近性为$t\searrow 0^+$。所得结果可用于分析与$Δ_ν$相关的谱ζ函数。
英文摘要：
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 $Δ_ν$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
