Asymptotics of resonances induced by point interactions

Abstract

We consider the resonances of the self-adjoint three-dimensional Schrödinger 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.

14 pages, 1 figure, submission to the proceedings of the 8th Workshop on Quantum Chaos and Localisation Phenomena, Warsaw, May 2017

References in corpus (2)

• On the Effective Size of a Non-Weyl Graph
• Pseudo orbit expansion for the resonance condition on quantum graphs and the resonance asymptotics

Cited by in corpus (2)

• On the multilevel internal structure of the asymptotic distribution of resonances
• Asymptotics of random resonances generated by a point process of delta-interactions