The Weyl symbol of Schr\"odinger semigroups

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.