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

In this paper we consider one-dimensional Schrodinger operators S(q)u(x) = -(-d(2)/dx(2) + q(r) (x) + q(s) (x)) u(x), x is an element of R, where q(r) is an element of L-infinity (R) is a regular potential with compact support, and q(s) is an element of D' (R) is a singular potential q(s) (x) = Sigma(N)(j=1)(alpha(j)delta(x - x(j)) + beta(j)delta'(x - x(j))), alpha(j), beta(j) is an element of C that 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 eta(+/-) that share the same structure up to an algebraic sign. The functions eta(+/-) 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) is an element of L-infinity (R) with compact support, even complexvalued, and point interactions with complex strengths alpha(j), beta(j) (j = 1,..., N). Moreover, our approach leads to an efficient numerical treatment of both the bound and resonance states. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

This paper investigates the characteristic functions of one-dimensional Schrödinger operators, discussing general properties and numerical treatment methods.