An optimal semiclassical bound on certain commutators

2019/12/18 by Søren Fournais, Fournais, Søren, Søren Mikkelsen +1
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1912.08467

openalex publication_date 2019/12/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove an optimal semiclassical bound on the trace norm of the following commutators [\boldsymbol1_(-∞,0](H_ℏ),x], [\boldsymbol1_(-∞,0](H_ℏ),-iℏ∇] and [\boldsymbol1_(-∞,0](H_ℏ),e^itx], where H_ℏ is a Schrödinger operator with a semiclassical parameter ℏ, x is the position operator and -iℏ∇ is the momentum operator. These bounds corresponds to a mean-field version of bounds introduced as an assumption by N. Benedikter, M. Porta and B. Schlein in a study of the mean-field evolution of a fermionic system.

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