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