  In this paper, we study the Weyl symbol of the Schr\"odinger semigroup
$e^{-tH}$, $H=-\Delta+V$, $t>0$, on $L^2(\mathbb{R}^n)$, with nonnegative
potentials $V$ in $L^1_{\rm loc}$. Some general estimates like the $L^{\infty}$
norm concerning the symbol $u$ are derived. In the case of large dimension,
typically for nearest neighbor or mean field interaction potentials, we prove
estimates with parameters independent of the dimension for the derivatives
$\partial_x^\alpha\partial_\xi^\beta u$. In particular, this implies that the
symbol of the Schr\"odinger semigroups belongs to the class of symbols
introduced in [1] in a high-dimensional setting. In addition, a commutator
estimate concerning the semigroup is proved.
