On the classification of the spectrally stable standing waves of the Hartree equation

We consider the fractional Hartree model, with general power non-linearity and arbitrary spatial 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.