  In this paper, we establish the uniqueness of positive solutions to the
following fractional nonlinear elliptic equation with harmonic potential
\begin{align*} (-\Delta)^s u+ \left(\omega+|x|^2\right) u=|u|^{p-2}u \quad
\mbox{in}\,\, \R^n, \end{align*} where $n \geq 1$, $0<s<1$,
$\omega>-\lambda_{1,s}$, $2<p<\frac{2n}{(n-2s)^+}$, and $\lambda_{1,s}>0$ is
the lowest eigenvalue of the operator $(-\Delta)^s + |x|^2$. This solves an
open question raised in \cite{SS} concerning the uniqueness of solutions to the
equation.
