Weyl composition of symbols in large dimension

TL;DR

This paper investigates the Weyl composition of symbols in high-dimensional spaces, providing dimension-independent estimates and analyzing the semiclassical expansion's remainder term for Weyl $h$-pseudodifferential operators.

Contribution

It introduces a class of symbols for uniform estimates of Weyl compositions in large dimensions, including regularized and hybrid compositions, with a focus on the remainder term analysis.

Findings

Dimension-independent bounds for Weyl symbol products

Decomposition formula for Weyl composition

Analysis of the semiclassical expansion remainder

Abstract

This paper is concerned with the Weyl composition of symbols in large dimension. We specify a class of symbols in order to estimate the Weyl symbol of the product of two Weyl $h-$pseudodifferential operators, with constants independent of the dimension. The proof includes a regularized and a hybrid compositions together with a decomposition formula. We also analyze in this context the remainder term of the semiclassical expansion of the Weyl composition.