The Weyl Symbol of Schrodinger Semigroups

In this paper, we study the Weyl symbol of the Schrodinger semigroup e(-tH) , H = -Delta + V, t > 0, on , with nonnegative potentials V in . Some general estimates like the L (a) 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 . In particular, this implies that the symbol of the Schrodinger semigroups belongs to the class of symbols introduced in Amour et al. (To appear in Proceedings of the AMS) in a high-dimensional setting. In addition, a commutator estimate concerning the semigroup is proved.