Schr\"odinger operators on the half-line with integrable complex potentials

TL;DR

This paper extends the spectral theory of half-line Schr"odinger operators with complex potentials by deriving explicit formulas for spectral pairs using Jost solutions, focusing on integrable potentials in L^1.

Contribution

It introduces explicit formulas for spectral pairs for non-self-adjoint Schr"odinger operators with integrable complex potentials, and analyzes associated Jost solutions.

Findings

Explicit spectral pair formulas in terms of Jost solutions

Existence and analysis of Jost solutions for L^1 potentials

Extension of spectral theory to complex integrable potentials

Abstract

In our previous work, we introduced the concept of a \emph{spectral pair} for a half-line Schr\"odinger 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 Schr\"odinger operator. A key component of our work, which is of independent interest, is the existence proof and analysis of these Jost solutions.