The Weyl Symbol of Schrödinger Semigroups

Abstract

In this paper, we study the Weyl symbol of the Schrodinger semigroup e−tH, H = −Δ + V, t > 0, on \({L^2(\mathbb{R}^n)}\) , with nonnegative potentials V in \({L^1_{\rm loc}}\) . Some general estimates like the L∞ 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 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.