The Weyl symbol of Schrödinger semigroups

Abstract

In this paper, we study the Weyl symbol of the Schrödinger semigroup $e^{-tH}$, $H=-\Delta +V$, $t>0$, on $L^2({\mathbb{R}}^n)$, with nonnegative potentials $V$ in $L_{\mathrm{loc}}^1$. 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 Schrödinger 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.

References in corpus (1)

• Weyl composition of symbols in large dimension

Cited by in corpus (1)

• Canonical mean-field molecular dynamics derived from quantum mechanics