Soren Mikkelsen

A semiclassical bound on certain commutators

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 = - ℏ²Δ+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.