  This paper is devoted to the $L^2$-constraint variational problem
\begin{equation*}
  We study $L^2$-normalized solutions of the following inhomogeneous fractional
Schr\"{o}dinger equation \begin{equation*} (-\Delta)^{s}
u(x)+V(x)u(x)-a|x|^{-b}|u|^{2\beta^2}u(x)=\mu u(x)\ \ \mbox{in}\ \ \R^{N}.
\end{equation*} Here $s\in(\frac{1}{2},1)$, $N>2s$, $a>0$,
$0<b<\min\{\frac{N}{2},1\}$, $\beta=\sqrt{\frac{2s-b}{N}}$ and $V(x)\geq 0$ is
an external potential. We get $L^2$-normalized solutions of the above equation
by solving the associated constrained minimization problem. We prove that there
exists a critical value $a^*>0$ such that minimizers exist for $0<a<a^*$, and
minimizers do not exist for any $a>a^*$. In the case of $a=a^*$, one can obtain
the classification results of the existence and non-existence for constraint
minimizers, which are depended strongly on the value of $V(0)$. For $V(0)=0$,
the limiting behavior of nonnegative minimizers is also analyzed when $a$ tend
to $a^*$ from below.
