论文标题
一般诱捕电势下非线性schrödinger方程的基态
Ground states for the nonlinear Schrödinger equation under a general trapping potential
论文作者
论文摘要
在过去的二十年中，对具有谐波陷阱电位$ v（x）= | x |^2 $的经典schrödinger方程进行了广泛的研究。其基础状态在局部积极解决方案中是钟形的，并且是独一无二的。此外，它们已被证明是非分类的，并且（强烈）轨道稳定。所有这些结果是在许多出版物和多个作者过程中产生的，依赖于专门为拉普拉斯式设计的ODE方法和功能函数潜力。 在本文中，我们对这些结果进行了广泛的概括。更具体地说，我们假设亚拉皮亚分数分散剂和非常通用的陷阱电位$ v $，驱动线性运算符的形式为$ h =（ - δ）^s+v，0 <s \ s \ leq 1 $。我们表明，存在这种半线性分数schrödinger方程的归一化波，如果非线性为$ | u | u |^{p-1} u，p <1+ \ frac {4 s} {n} $。此外，我们表明这样的波是非分化的，并且轨道稳定。即使在经典案例$ h =-Δ+v $中，这些结果还是新的，其中$ v $是本文中考虑的一般陷阱潜力。
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.
