Asymptotics of resonances induced by point interactions

We consider the resonances of the self-adjoint three-dimensional Schr\"odinger operator with point interactions of constant strength supported on the set $X = \{ x_n \}_{n=1}^N$. The size of $X$ is defined by $V_X = \max_{\pi\in\Pi_N} \sum_{n=1}^N |x_n - x_{\pi(n)}|$, where $\Pi_N$ is the family of all the permutations of the set $\{1,2,\dots,N\}$. We prove that the number of resonances counted with multiplicities and lying inside the disc of radius $R$ behaves asymptotically linear $\frac{W_X}{\pi} R + \mathcal{O}(1)$ as $R \to \infty$, where the constant $W_X \in [0,V_X]$ can be seen as the effective size of $X$. Moreover, we show that there exist configurations of any number of points such that $W_X = V_X$. Finally, we construct an example for $N = 4$ with $W_X < V_X$, which can be viewed as an analogue of a quantum graph with non-Weyl asymptotics of resonances.