A power series analysis of bound and resonance states of one-dimensional Schrödinger operators with finite point interactions

Abstract

In this paper we consider one-dimensional Schrödinger operatorsSqu(x)=(−d2dx2+qr(x)+qs(x))u(x),x∈R,where qr∈L∞(R) is a regular potential with compact support, and qs∈D′(R) is a singular potentialqs(x)=∑j=1N(αjδ(x−xj)+βjδ′(x−xj)),αj,βj∈Cthat involves a finite number of point interactions. The eigenenergies associated to the bound states and the complex energies associated to the resonance states of operator Sq are given by the zeros of certain characteristic functions η± that share the same structure up to an algebraic sign. The functions η± are obtained explicitly in the form of power series of the spectral parameter, and the computation of the coefficients of the series is given by a recursive integration procedure. The results here presented are general enough to consider arbitrary regular potentials qr∈L∞(R) with compact support, even complex-valued, and point interactions with complex strengths αj,βj (j=1,…,N). Moreover, our approach leads to an efficient numerical treatment of both the bound and resonance states.