On Scattering of Poles for Schr\"{o}dinger Operator from the Point of View of Dirichlet Series

Abstract:

      In this article, we are concerned with the scattering problem of Schr\"{o}dinger operators with compactly supported potentials on the real line. We aim at combining the theory of Dirichlet series with scattering theory. New estimate on the number of poles is obtained under the situation that the growth of power series which is related to the potential is not too fast by using a classical result of Littlewood. We propose a new approach of Dirichlet series such that significant upper bounds and lower bounds on the number of poles are obtained. The results obtained in this paper improve and extend some related conclusions on this topic.