  The Kato square root problem for divergence form elliptic operators with
potential $V : \mathbb{R}^{n} \rightarrow \mathbb{C}$ is the equivalence
statement $\left\Vert (L + V)^{\frac{1}{2}} u\right\Vert_{2} \simeq \left\Vert
\nabla u \right\Vert_{2} + \left\Vert V^{\frac{1}{2}} u \right\Vert_{2}$, where
$L + V := - \mathrm{div} A \nabla + V$ and the perturbation $A$ is an
$L^{\infty}$ complex matrix-valued function satisfying an accretivity
condition. This relation is proved for any potential with range contained in
some positive sector and satisfying $\left\Vert |V|^{\frac{\alpha}{2}}
u\right\Vert_{2} + \left\Vert (-\Delta)^{\frac{\alpha}{2}} \right\Vert_{2}
\lesssim \left\Vert ( |V| - \Delta)^{\frac{\alpha}{2}}u \right\Vert_{2}$ for
all $u \in D(|V| -\Delta)$ and some $\alpha \in (1,2]$. The class of potentials
that will satisfy such a condition is known to contain the reverse H\"{o}lder
class $RH_{2}$ and $L^{\frac{n}{2}}(\mathbb{R}^{n})$ in dimension $n > 4$. To
prove the Kato estimate with potential, a non-homogeneous version of the
framework introduced by A. Axelsson, S. Keith and A. McIntosh for proving
quadratic estimates is developed. In addition to applying this non-homogeneous
framework to the scalar Kato problem with zero-order potential, it will also be
applied to the Kato problem for systems of equations with zero-order potential.
