  We establish sharp pointwise estimates for the ground states of some singular
fractional Schr\"odinger operators on relatively compact Euclidean subsets. The
considered operators are of the type $(-\Delta)^{\alpha/2}|_\Omega-V$, where
$V\in L^1_{Loc}$ and $(-\Delta)^{\alpha/2}|_\Omega$ is the fractional-Laplacian
on an open subset $\Omega$ in $\mathbb{R}^d$ with zero exterior condition . The
intrinsic ultracontractivity property for such operators is discussed as well
and a sharp large time asymptotic for their heat kernels is derived.
