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Ground states for the nonlinear Schrödinger equation under a general trapping potential

The classical Schrödinger equation with a harmonic trap potential $V(x)=|x|^2$, describing the quantum harmonic oscillator, has been studied quite extensively in the last twenty years. Its ground states are bell-shaped and unique, among localized positive solutions. In addition, they have been shown to be non-degenerate and (strongly) orbitally stable. All of these results, produced over the course of many publications and multiple authors, rely on ODE methods specifically designed for the Laplacian and the power function potential. In this article, we provide a wide generalization of these results. More specifically, we assume sub-Laplacian fractional dispersion and a very general form of the trapping potential $V$, with the driving linear operator in the form $H=(-Δ)^s+V, 0<s\leq 1$. We show that the normalized waves of such semi-linear fractional Schrödinger equation exist, they are bell-shaped, provided that the non-linearity is of the form $|u|^{p-1} u, p<1+\frac{4 s}{n}$. In addition, we show that such waves are non-degenerate, and strongly orbitally stable. Most of these results are new even in the classical case $H=-Δ+V$, where $V$ is a general trapping potential considered herein.