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

In this paper we consider one-dimensional Schrödinger operators [Formula presented] where q_r∈L^∞(R) is a regular potential with compact support, and q_s∈D^′(R) is a singular potential q_s(x)=∑j=1N(α_jδ(x−x_j)+β_jδ^′(x−x_j)),α_j,β_j∈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 S_q 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 q_r∈L^∞(R) with compact support, even complex-valued, and point interactions with complex strengths α_j,β_j (j=1,…,N). Moreover, our approach leads to an efficient numerical treatment of both the bound and resonance states.