We determine the principal term of the asymptotics of the integrated density of states (IDS) \(N(\lambda )\) for the Schrödinger operator with point interactions on \({\textbf{R}}^3\) as \(\lambda \rightarrow -\infty ,\) 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 \(\lambda \rightarrow -\infty .\) 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 the R programming language. As a byproduct, we give a rigorous definition of the Schrödinger operators with point interactions on a bounded open set with the Dirichlet or Neumann boundary conditions by the method of quadratic form, and study fundamental properties about the counting functions of eigenvalues; e.g., Dirichlet–Neumann bracketing, etc.
