$L^p$-regularity for fourth-order Schrödinger operators with unbounded or degenerate coefficients

We prove that operators of the form $A=-{\mathrm{\Delta }}^2-V(x)$, with $\mathrm{\mid }DV(x)\mathrm{\mid }\le \gamma V(x{)}^{\frac{5}{4}}+C_{\gamma }$ for some $\gamma$ small enough and $C_{\gamma }$ positive constant, generate analytic semigroups in $L^p({\mathbb{R}}^N)$ for $1\le p<\mathrm{\infty }$ and in $C_0({\mathbb{R}}^N)$. Moreover, we characterize the maximal domain of such operators.