Generic asymptotics of resonance counting function for Schr\"odinger point interactions

TL;DR

This paper proves that the typical asymptotic behavior of the resonance counting function for Schrödinger operators with point interactions follows a Weyl-type law, establishing a fundamental property of these quantum systems.

Contribution

It demonstrates that Weyl-type asymptotics for the resonance counting function is a generic feature for Schrödinger Hamiltonians with point interactions.

Findings

Weyl-type asymptotics is shown to be generic.

The leading coefficient in the asymptotic formula is characterized.

Non-Weyl-type asymptotics are not typical.

Abstract

We study the leading coefficient in the asymptotical formula $ N (R) = \frac{W}{\pi} R + O (1) $, $ R \to \infty $, for the resonance counting function $ N (R)$ of Schr\"odinger Hamiltonians with point interactions. For such Hamiltonians, the Weyl-type and non-Weyl-type asymptotics of $N (R)$ was introduced recently in a paper by J. Lipovsk\'y and V. Lotoreichik (2017). In this note, we prove that the Weyl-type asymptotics is generic.