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

Abstract

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.