  We prove that the operator $L_0=-(1+|x|)^\beta(-\Delta)^{\alpha/2}$ with
$\alpha\in(0,2)$, $d>\alpha$ and $\beta\ge0$ generates a compact semigroup or
resolvent on $L^2(\R^d;(1+|x|)^{-\beta}\,dx)$, if and only if $\beta>\alpha$.
When $\beta>\alpha$, we obtain two-sided asymptotic estimates for high order
eigenvalues, and sharp bounds for the corresponding heat kernel.
