Commutator Estimates and Quantitative Local Weyl's Law for Schrödinger Operators with Non-Smooth Potentials

2025/01/02 by Cárdenas, Esteban, Lafleche, Laurent · 2 citations
#35J10 #35P30 (Secondary) #47B10 #47B47 (Primary) 47B15 #81S30 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Physics (quant-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2501.01381

Abstract

We analyze semi-classical Schrödinger operators with potentials of class C^1,1/2 and establish commutator estimates for the associated projection operators in Schatten norms. These are then applied to prove quantitative versions of the local and phase space Weyl laws in L^p spaces. We study both non-interacting, and interacting particle systems. In particular, we are able to treat the case of the minimizers of the Hartree energy in the case of repulsive singular pair interactions such as the Coulomb potential.

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• Ground state energy of a dilute inhomogeneous Fermi gas
• The quantitative semi-classical limit of a large Fermi system at zero temperature

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