The Weyl Symbol of Schrödinger Semigroups

In this paper, we study the Weyl symbol of the Schrödinger semigroup e−tH, H = −Δ + V, t > 0, on L2(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^2(\mathbb{R}^n)}$$\end{document} , with nonnegative potentials V in Lloc1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^1_{\rm loc}}$$\end{document} . 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 ∂xα∂ξβu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\partial_x^\alpha\partial_\xi^\beta u}$$\end{document} . In particular, this implies that the symbol of the Schrödinger 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.