Spectral decomposition for the 1-D Schroedinger-Poisson equation

Abstract

In this article we are concerned with the 1-D Schr ¨odinger equation i ut = −uxx − V (u) u , where V (u) is a Hartree nonlinearity, stemming from the coupling with the Poisson equation. Using the Titchmarsh-Kodaira’s theorem, we give an explicit spectral decomposition of the related linear operator uxx + |x| u and we show that the evolution consists only of scattering states.