# American Institute of Mathematical Sciences

doi: 10.3934/era.2021047
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## On a fractional Schrödinger equation in the presence of harmonic potential

 1 Mathematics Department, University of Wisconsin-Madison, Madison, WI 53706, USA 2 Department of Mathematics, California State University, Los Angeles, Los Angeles, CA 90032, USA

* Corresponding author: Hichem Hajaiej

Received  November 2020 Revised  March 2021 Early access June 2021

In this paper, we establish the existence of ground state solutions for a fractional Schrödinger equation in the presence of a harmonic trapping potential. We also address the orbital stability of standing waves. Additionally, we provide interesting numerical results about the dynamics and compare them with other types of Schrödinger equations [11,18]. Our results explain the effect of each term of the Schrödinger equation: the fractional power, the power of the nonlinearity, and the harmonic potential.

Citation: Zhiyan Ding, Hichem Hajaiej. On a fractional Schrödinger equation in the presence of harmonic potential. Electronic Research Archive, doi: 10.3934/era.2021047
##### References:

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##### References:
Ground state solutions with $\sigma = 1$ and different $s$
Ground state solutions with non-symmetric potential
Energy and stability check with $s = 0.8$, $\sigma = 1$
Abosolute value of solution with different $s$
Stability test with $\psi^s_0(x) = 0.9*u^s_0(x)$
Ground state solution with $s = 0.8$, $\sigma = 1$, $L = 10$ and $J = 5000$
Time dynamics of standing waves with $s = 0.8$, $\sigma = 1$, $L = 10$ and $J = 5000$
$L^2$ distance between ground state solutions of $s<1$ and $s = 1$ with $\sigma = 1$
Energy and $\lambda_c$
$\|\psi^s(x, t)-u^s(x, t)\|_{\widetilde{\Sigma}_s(\mathbb{R})}$ when $s = 0.8$ and $\sigma = 1$
Standing wave, ground state solution and blow up solution with $s = 0.5$, $\delta = 1$
Dynamics of FNLS
Evolution of $L^\infty$
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