### A semiclassical bound on certain commutators

Analysis and Geometry Seminar

31st October 2019, 3:15 pm – 4:15 pm

Fry Building, 2.04

In a series of papers N. Benedikter, M. Porta and B. Schlein considered time evolution of systems of elementary particles (fermions). In these studies they were able to describe how the particles evolve in time in certain settings given some initial state of the system. One assumption on the initial state was concerning a semiclassical bound on certain commutators. A mean-field version of this bound takes the form

|| [ *A*, 1_{(-∞,0]}(*H _{ℏ}*) ] ||

_{1}≤

*C ℏ*,

^{1-d}where *H _{ℏ} = - ℏ^{2}Δ+V* is a Schrödinger operator,

*A*is either the position operator

*X*or the momentum operator

*−iℏ∇*,

*C*is a positive constant and ||·||

_{1}denotes the trace norm. In this talk we will discuss ideas and methods used in a proof of such semiclassical commutator bounds.

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