Asymptotics of Random Resonances Generated by a Point Process of Delta-Interactions

Abstract

We introduce and study the following model for random resonances: we take a collection of point interactions $\Upsilon_j$ generated by a simple finite point process in the 3-D space and consider the resonances of associated random Schr\"odinger Hamiltonians $H_\Upsilon = -\Delta + ``\sum \mathfrak{m}(\alpha) \delta (x - \Upsilon_j)``$. These resonances are zeroes of a random exponential polynomial, and so form a point process $\Sigma (H_\Upsilon)$ in the complex plane $\mathbb{C}$. We show that the counting function for the set of random resonances $\Sigma (H_\Upsilon)$ in $\mathbb{C}$-discs with growing radii possesses Weyl-type asymptotics almost surely for a uniform binomial process $\Upsilon$, and obtain an explicit formula for the limiting distribution as $m \to \infty$ of the leading parameter of the asymptotic chain of `most narrow' resonances generated by a sequence of uniform binomial processes $\Upsilon^m$ with $m$ points. We also pose a general question about the limiting behavior of the point process formed by leading parameters of asymptotic sequences of resonances. Our study leads to questions about metric characteristics for the combinatorial geometry of $m$ samples of a random point in the 3-D space and related statistics of extreme values.