$L^p$ Maximal regularity for vector-valued Schrödinger operators

In this paper we consider the vector-valued Schrödinger operator $-\mathrm{\Delta }+V$, where the potential term $V$ is a matrix-valued function whose entries belong to $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^d)$ and, for every $x\in {\mathbb{R}}^d$, $V(x)$ is a symmetric and nonnegative definite matrix, with non positive off-diagonal terms and with eigenvalues comparable each other. For this class of potential terms we obtain maximal inequality in $L^1({\mathbb{R}}^d,{\mathbb{R}}^m)\mathrm{.}$ Assuming further that the minimal eigenvalue of $V$ belongs to some reverse Hölder class of order $q\in (1,\mathrm{\infty })\cup \{\mathrm{\infty }\}$, we obtain maximal inequality in $L^p({\mathbb{R}}^d,{\mathbb{R}}^m)$, for $p$ in between $1$ and some $q$.