# On the classification of the spectrally stable standing waves of the Hartree problem

## TL;DR

This paper constructs and classifies spectrally stable standing waves in the fractional Hartree and classical Hartree models, establishing their properties and stability conditions across various dimensions and nonlinearities.

## Contribution

It provides a variational construction of normalized solutions for the fractional Hartree model and classifies all spectrally stable solitons in the classical Hartree problem.

## Key findings

- Normalized solutions are spectrally stable in the fractional Hartree model.
- Only normalized solutions are spectrally stable in the classical Hartree problem.
- Spectral stability depends on the existence range of normalized solutions.

## Abstract

We consider the fractional Hartree model, with general power non-linearity and space dimension. We construct variationally the "normalized" solutions for the corresponding Choquard-Pekar model - in particular a number of key properties, like smoothness and bell-shapedness are established. As a consequence of the construction, we show that these solitons are spectrally stable as solutions to the time-dependent Hartree model. In addition, we analyze the spectral stability of the Moroz-Van Schaftingen solitons of the classical Hartree problem, in any dimensions and power non-linearity. A full classification is obtained, the main conclusion of which is that only and exactly the "normalized" solutions (which exist only in a portion of the range) are spectrally stable.