The Weyl symbol of Schr\"odinger semigroups

TL;DR

This paper analyzes the Weyl symbol of Schr"odinger semigroups, providing estimates on its behavior and derivatives, especially in high dimensions, and establishing its membership in a specific symbol class.

Contribution

It introduces dimension-independent estimates for the Weyl symbol of Schr"odinger semigroups with nonnegative potentials, extending understanding in high-dimensional analysis.

Findings

Derived $L^{ abla}$ norm estimates for the Weyl symbol.

Proved dimension-independent derivative estimates for the symbol.

Established the symbol's membership in a specific high-dimensional symbol class.

Abstract

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.