$L^p$-regularity for fourth-order Schr\"odinger operators with unbounded or degenerate coefficients

Abstract

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