 1784 Jiri Lipovsky, Vladimir Lotoreichik
 Asymptotics of resonances induced by point interactions
(692K, pdf)
Aug 11, 17

Abstract ,
Paper (src),
View paper
(auto. generated pdf),
Index
of related papers

Abstract. We consider the resonances of the selfadjoint threedimensional 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 $R$ asymptotically behaves as $rac{W_X}{\pi} R + \mathcal{O}(1)$ as $R o \infty$, where $W_X \in [0,V_X]$ is 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 nonWeyl quantum graph.
 Files:
1784.src(
1784.comments ,
1784.keywords ,
point_interactions.pdf.mm )