The Schrödinger operator $T = (i\nabla +b)^2+a \cdot \nabla + q$ on $\mathbb{R}^N$ is considered for $N \ge 2$. Here $a=(a_{j})$ and $b=(b_{j})$ are real-vector-valued functions on $\mathbb{R}^N$, while $q$ is a complex-scalar-valued function on $\mathbb{R}^N$. Over twenty years ago late Professor Kato proved that the minimal realization $T_{min}$ is essentially quasi-$m$-accretive in $L^2(\mathbb{R}^N)$ if, among others, $(1+|x|)^{-1}a_j \in L^4(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)$. In this paper it is shown that under some additional conditions the same conclusion remains true even if $a_j \in L^4_{loc}(\mathbb{R}^N)$.