arXiv 1801.075982018-01-22math.SPmath-phmath.MP

椭圆算子的加权局部Weyl律

Weighted local Weyl laws for elliptic operators

Alejandro Rivera

备注：: major changes and corrections; 46 pages. arXiv admin note: text overlap with arXiv:1611.02018

摘要：

令$A$是维度为$n>0$的闭流形$\mathcal{X}$上的$m$阶椭圆伪微分算子，关于某个正光滑密度$dμ_\mathcal{X}$形式上为正自伴。然后，$A$的谱由特征值序列$(λ_k)_{k\geq 1}$组成，其对应的特征函数$(e_k)_{k\geq 1}$是$C^\infty$光滑的。固定$s\in\mathbb{R}$并定义\[ K_L^s(x,y)=\sum_{0<λ_k\leq L}λ_k^{-s} e_k(x)\overline{e_k(y)}\, .\]我们推导出当$L\rightarrow +\infty$具有固定的$s$时，核$K_L^s(x,y)$在对角线附近的渐近公式。对于$s=0$，$K^0_L$是Hörmander在\cite{ho68}中研究的光谱投影仪的核心。在目前的工作中，我们建立在Hörmander的结果来研究内核$K^s_L$。如果$s<\frac{n}{m}$，$K_L^s$的阶数为$L^{-s+n/m}$并且在对角线附近，则重新缩放的前导项的行为类似于$A$的符号的显式函数的傅立叶变换。如果$s=\frac{n}{m}$，在$A$的主符号上的某些显式一般条件下，如果$A$是微分算子，则该条件成立，核具有$\ln(L)$阶，并且首项具有在$L^{-1/m}$尺度上平滑的对数发散。我们的结果也适用于椭圆型微分Dirichlet特征值问题。

英文摘要：

Let $A$ be an elliptic pseudo-differential operator of order $m$ on a closed manifold $\mathcal{X}$ of dimension $n>0$, formally positive self-adjoint with respect to some positive smooth density $dμ_\mathcal{X}$. Then, the spectrum of $A$ is made up of a sequence of eigenvalues $(λ_k)_{k\geq 1}$ whose corresponding eigenfunctions $(e_k)_{k\geq 1}$ are $C^\infty$ smooth. Fix $s\in\mathbb{R}$ and define \[ K_L^s(x,y)=\sum_{0<λ_k\leq L}λ_k^{-s} e_k(x)\overline{e_k(y)}\, .\] We derive asymptotic formulae near the diagonal for the kernels $K_L^s(x,y)$ when $L\rightarrow +\infty$ with fixed $s$. For $s=0$, $K^0_L$ is the kernel of the spectral projector studied by Hörmander in \cite{ho68}. In the present work we build on Hörmander's result to study the kernels $K^s_L$. If $s<\frac{n}{m}$, $K_L^s$ is of order $L^{-s+n/m}$ and near the diagonal, the rescaled leading term behaves like the Fourier transform of an explicit function of the symbol of $A$. If $s=\frac{n}{m}$, under some explicit generic condition on the principal symbol of $A$, which holds if $A$ is a differential operator, the kernel has order $\ln(L)$ and the leading term has a logarithmic divergence smoothed at scale $L^{-1/m}$. Our results also hold for elliptic differential Dirichlet eigenvalue problems.

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