Schrödinger operators on the half-line with integrable complex potentials

In our previous work, we introduced the concept of a \emph{spectral pair} for a half-line Schrödinger operator with a \emph{complex} bounded potential $q$, serving as a substitute for the spectral measure in a non-self-adjoint setting. In this paper, we study the case of $q\in L^1({\mathbb{R}}_{+})$. We derive explicit formulas for the spectral pair in terms of the Jost solutions of a system of two equations naturally associated with the non-self-adjoint Schrödinger operator. A key component of our work, which is of independent interest, is the existence proof and analysis of these Jost solutions.