Scale invariant elliptic operators with singular coefficients

We show that a realization of the operator $L=|x|^\alpha\Delta +c|x|^{\alpha-1}\frac{x}{|x|}\cdot\nabla -b|x|^{\alpha-2}$ generates a semigroup in $L^p({\mathbb{R}}^N)$ if and only if $D_c=b+(N-2+c{)}^2\mathrm{/}4>0$ and $s_1+\min \{0,2-\alpha \}<N\mathrm{/}p<s_2+\max \{0,2-\alpha \}$, where $s_i$ are the roots of the equation $b+s(N-2+c-s)=0$, or $D_c=0$ and $s_0+\min{0,2-\alpha} \le N/p \le s_0+\max{0,2-\alpha}$,where$s_0$ is the unique root of the above equation. The domain of the generator is also characterized.