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Fermionic semiclassical Lp estimates

Ngoc Nhi Nguyen

TL;DR

This work extends the semiclassical $L^p$ density estimates originally developed for Schrödinger operators on compact manifolds to density matrices, enabling analysis of fermionic systems in trapping potentials. By combining microlocal analysis with many-body tools, it establishes elliptic, Sogge-type, and turning-point $L^p$ bounds for density matrices and their associated particle densities, with exponents that depend on dimension and spectral-region geometry. The results provide sharp one-body and many-body bounds, including spectral-cluster bounds, and illuminate the transition between localization and delocalization via multi-scale turning-point analysis around the energy surface $\{V=E\}$. The framework handles microlocalized quasimodes and density matrices, offering precise Schatten-norm control and dispersive-estimate-based proofs, with explicit endpoint analyses and turning-point scaling $h^{2/3}$ that capture concentration phenomena in trapped fermionic systems.

Abstract

We generalize the semiclassical Lp estimates of Koch, Tataru and Zworski in the setting of Schr{ö}dinger operators with confining potentials to density matrices. This is motivated by the problem of the concentration of free fermionic particles in a trapping potential. Our proof relies on semiclassical and many-body tools. As an application, we provide bounds on spectral clusters. We also discuss the optimality of the one-body and many-body bounds through explicit examples of quasimodes.

Fermionic semiclassical Lp estimates

TL;DR

This work extends the semiclassical density estimates originally developed for Schrödinger operators on compact manifolds to density matrices, enabling analysis of fermionic systems in trapping potentials. By combining microlocal analysis with many-body tools, it establishes elliptic, Sogge-type, and turning-point bounds for density matrices and their associated particle densities, with exponents that depend on dimension and spectral-region geometry. The results provide sharp one-body and many-body bounds, including spectral-cluster bounds, and illuminate the transition between localization and delocalization via multi-scale turning-point analysis around the energy surface . The framework handles microlocalized quasimodes and density matrices, offering precise Schatten-norm control and dispersive-estimate-based proofs, with explicit endpoint analyses and turning-point scaling that capture concentration phenomena in trapped fermionic systems.

Abstract

We generalize the semiclassical Lp estimates of Koch, Tataru and Zworski in the setting of Schr{ö}dinger operators with confining potentials to density matrices. This is motivated by the problem of the concentration of free fermionic particles in a trapping potential. Our proof relies on semiclassical and many-body tools. As an application, we provide bounds on spectral clusters. We also discuss the optimality of the one-body and many-body bounds through explicit examples of quasimodes.
Paper Structure (54 sections, 49 theorems, 386 equations, 14 figures, 5 tables)

This paper contains 54 sections, 49 theorems, 386 equations, 14 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Review of semiclassical analysis and density matrices
  3. Symbol classes and quantization
  4. Semiclassical pseudodifferential calculus
  5. Semiclassical bounds
  6. Density matrices
  7. Strichartz estimates for density matrices
  8. Step 1. Operator norm bounds.
  9. Step 2. Schatten $\mathfrak{S}^2$-bounds.
  10. Step 3. Conclusion.
  11. Relations between various estimates on quasimodes
  12. Elliptic estimates
  13. More general L$^p$ estimates
  14. Statement of the result
  15. Notation for the proof of Theorem \ref{['thm:ELp-gene-matdens']}
  16. ...and 39 more sections

Key Result

Theorem 1

Let the symbol $p(x,\xi)=|\xi|^2+V(x)$ with a confiningto be defined in Section sec:app-spectral-clusters potential $V:\mathbb{R}^d\to\mathbb{R}$, $E\in\mathbb{R}$, $\varepsilon>0$ be a small error. Let us denote by $P$ by the Schrödinger operator $-h^2\Delta+V$ and by $\Pi_{E,h}$ the spectral proje Assume that $\Omega=\mathbb{R}^d$, or $\Omega=\{V>E+\varepsilon\}$ in the classical forbidden regio

Figures (14)

  • Figure 1: Eigenfunction of the scalar harmonic oscillator associated to the eigenvalue $E$.
  • Figure 2: Spectral projector of the scalar harmonic oscillator.
  • Figure 3: Exponent $s(q,d)$ of microlocalized estimates when $d\geq 2$.
  • Figure 4: Schatten exponent $\alpha(q,d)$ when $d\geq 3$.
  • Figure 5: Exponent $s(q,d)$ and $\alpha(q,d)$ for elliptic estimates.
  • ...and 9 more figures

Theorems & Definitions (115)

  • Theorem 1: Spectral cluster $L^q$ estimates, see Theorem \ref{['thm:spectral-clusters']} and Section \ref{['subsec:optim-manybody']}
  • Theorem 2: Microlocalized $L^q$ estimates
  • Definition II.1: Order functions zworski2012semiclassical
  • Remark 1
  • Definition II.2: Symbols zworski2012semiclassical
  • Remark 2
  • Remark 3
  • Definition II.3
  • Remark 4
  • Definition II.4: Boundedness from below
  • ...and 105 more