标题： $\mathbf{R}^3$上泊松点相互作用的态密度 显示英文标题

标题： Integrated density of states for the Poisson point interactions on $\mathbf{R}^3$

摘要： 我们确定了在点障碍的位置为泊松配置，且相互作用参数为有界独立同分布随机变量时，薛定谔算子在$\mathbf{R}^3$上的积分态密度（IDS）$N(\lambda)$的渐近主项，当$\lambda \to -\infty$时。 特别是，我们证明了$N(\lambda) =O(|\lambda|^{-3/2})$当$\lambda\to -\infty$时成立。 当所有相互作用参数等于常数时，我们给出了$N(\lambda)$更详细的渐近形式，并通过使用 R 的数值方法验证了该结果。 显示英文摘要

摘要： We determine the principal term of the asymptotics of the integrated density of states (IDS) $N(\lambda)$ for the Schr\"odinger operator with point interactions on $\mathbf{R}^3$ as $\lambda \to -\infty$, provided that the set of positions of the point obstacles is the Poisson configuration, and the interaction parameters are bounded i.i.d.\ random variables. In particular, we prove $N(\lambda) =O(|\lambda|^{-3/2})$ as $\lambda\to -\infty$. In the case that all interaction parameters are equal to a constant, we give a more detailed asymptotics of $N(\lambda)$, and verify the result by a numerical method using R.