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 20 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 \({{\mathscr {H}}}=(-\Delta )^s+V, 0<s\le 1\). We show that the normalized waves of such semilinear fractional Schrödinger equation exist, and they are bell-shaped, provided that the nonlinearity 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 \({{\mathscr {H}}}=-\Delta +V\), where V is a general trapping potential considered herein.

在最近的20年中，对描述谐波谐振子的具有势阱陷阱势\（V（x）= | x | ^ 2 \）的经典Schrödinger方程进行了广泛的研究。在局部正解中，其基态为钟形且独特。此外，它们已被证明是非简并和（强烈）轨道稳定的。所有这些结果，是在许多出版物和多位作者的过程中得出的，都依赖于专门为拉普拉斯算术和幂函数潜力设计的ODE方法。在本文中，我们对这些结果进行了广泛的概括。更具体地说，我们假设亚拉普拉斯分数色散和俘获势V的非常一般形式，而驱动线性算子的形式为\（{{\ mathscr {H}}} =（-Delta ^ s + V，0 <s \ le 1 \）。我们证明存在这种半线性分数Schrödinger方程的归一化波，并且它们呈钟形，只要非线性形式为\（| u | ^ {p-1} u，p <1+ \ frac {4 s} {n} \）。另外，我们证明了这些波是非简并的并且在轨道上很稳定。即使在经典情况\（{{\ mathscr {H}}} =-\ Delta + V \）中，这些结果大多数都是新的，其中V是此处考虑的一般捕获势。