实直线上的测度几何拉普拉斯算子
Measure-geometric Laplacians on the real line
摘要：
受微积分基本定理的启发，在Feller以及Kac和Kre \u n的工作基础上，给出了$\mathbb{R}$的紧子集上支持的无原子Borel概率测度$η$，Freiberg和Zähle引入了一种测度几何方法来定义关于$η$的一阶微分算子$\nabla_η$和二阶微分算子$Δ_η$.我们将这种方法推广到$η= ν+ δ$形式的措施，其中$ν$是连续的和$δ$是连续的支持。我们确定了$\nabla_η$和$Δ_η$的解析性质，并证明了$Δ_η$是一个稠密的、无界的、线性的、具有紧预解式的自伴算子.此外，我们还给出了计算$Δ_η$特征值和特征函数的系统方法。对于两个主要例子，我们确定了特征值和特征函数，以及特征值计数函数的渐进增长率。
英文摘要：
Motivated by the fundamental theorem of calculus, and based on the works of Feller as well as Kac and Kre\uın, given an atomless Borel probability measure $η$ supported on a compact subset of $\mathbb{R}$, Freiberg and Zähle introduced a measure-geometric approach to define a first order differential operator $\nabla_η$ and a second order differential operator $Δ_η$, with respect to $η$. We generalise this approach to measures of the form $η= ν+ δ$, where $ν$ is continuous and $δ$ is finitely supported. We determine analytic properties of $\nabla_η$ and $Δ_η$ and show that $Δ_η$ is a densely defined, unbounded, linear, self-adjoint operator with compact resolvent. Moreover, we give a systematic way to calculate the eigenvalues and eigenfunctions of $Δ_η$. For two leading examples, we determine the eigenvalues and the eigenfunctions, as well as the asymptotic growth rates of the eigenvalue counting function.
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