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

Esteban Cárdenas, Laurent Lafleche

TL;DR

This work develops sharp commutator estimates for semiclassical Schrödinger operators with non-smooth potentials and uses them to derive quantitative local and phase-space Weyl laws for both non-interacting and interacting (Hartree) particle systems. The authors establish uniform-in-$\hbar$ Schatten-norm bounds for commutators $[x,\boldsymbol{\gamma}]$ and $[\hbar\nabla,\boldsymbol{\gamma}]$, relate these to regularity of $f_{\boldsymbol{\gamma}}$ and $\varrho_{\boldsymbol{\gamma}}$, and prove Besov-type phase-space bounds and Agmon-type decay. They extend the linear theory to the Hartree nonlinear setting, obtaining nonlinear commutator bounds that depend on the singularity of the interaction $K(x)=\kappa|x|^{-a}$, including the Coulomb case ($a=1$) with logarithmic corrections. The results yield precise rates of convergence for densities and Wigner functions to Thomas–Fermi limits, and provide robust tools for semiclassical analysis of many-body quantum systems with non-smooth interactions. Collectively, these results strengthen the quantum-classical correspondence in non-smooth regimes and have potential implications for effective mean-field descriptions of interacting fermions, especially in Coulombic environments.

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.

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

TL;DR

This work develops sharp commutator estimates for semiclassical Schrödinger operators with non-smooth potentials and uses them to derive quantitative local and phase-space Weyl laws for both non-interacting and interacting (Hartree) particle systems. The authors establish uniform-in- Schatten-norm bounds for commutators and , relate these to regularity of and , and prove Besov-type phase-space bounds and Agmon-type decay. They extend the linear theory to the Hartree nonlinear setting, obtaining nonlinear commutator bounds that depend on the singularity of the interaction , including the Coulomb case () with logarithmic corrections. The results yield precise rates of convergence for densities and Wigner functions to Thomas–Fermi limits, and provide robust tools for semiclassical analysis of many-body quantum systems with non-smooth interactions. Collectively, these results strengthen the quantum-classical correspondence in non-smooth regimes and have potential implications for effective mean-field descriptions of interacting fermions, especially in Coulombic environments.

Abstract

We analyze semi-classical Schrödinger operators with potentials of class 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 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.
Paper Structure (26 sections, 42 theorems, 376 equations)

This paper contains 26 sections, 42 theorems, 376 equations.

Table of Contents

  1. Introduction
  2. Weyl's laws
  3. Commutator estimates and semiclassical regularity
  4. Interacting particles
  5. Main results
  6. Commutator estimates and Weyl's law
  7. The interacting case
  8. Strategy of proof
  9. A priori estimates
  10. Lp estimates
  11. Energy estimates
  12. Agmon-like estimates
  13. Commutator estimates
  14. Motivation
  15. Eigenvalue estimates
  16. ...and 11 more sections

Key Result

Theorem 1.1

For $\hbar\in(0,1)$, let ${\boldsymbol{\gamma}} = \mathds{1}_{H \leq 0}$ where $H = -\hbar^2 \Delta + V$ in $d = 3$. Assume that for some $\beta \geq 0$ uniformly in $\hbar$. Then, for any $p\in [1,\infty)$, there exists a constant $C>0$ independent of $\hbar$ such that Additionally, if $e^{-\beta\left|{x}\right|}\,\nabla^2 V\in L^\infty(\mathbb R^3)$ uniformly in $\hbar$, then the $\left|{\ln \h

Theorems & Definitions (100)

  • Theorem 1.1
  • Remark 1.4.1
  • Remark 1.4.2
  • Remark 1.4.3
  • Corollary 1.2: Regularity of the density
  • Theorem 1.3
  • Remark 1.4.4
  • Theorem 1.4: Linear quantitative convergence
  • Remark 1.4.5
  • Remark 1.4.6
  • ...and 90 more