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Lieb--Thirring inequalities for large quantum systems with inverse nearest-neighbor interactions

G. K. Duong, Phan Thành Nam

TL;DR

This work extends Lieb–Thirring-type energy bounds to many-body quantum systems without antisymmetry by replacing the standard two-body repulsion with an inverse nearest-neighbor interaction and employing a fractional kinetic energy $(-\Delta)^s$. The authors develop a microlocal localization framework, including local uncertainty and local exclusion principles built on a Besicovitch-based multiscale covering, to derive global density-based bounds. They establish a Lieb–Thirring inequality with the inverse-nearest-neighbor potential, prove a Hardy–Lieb–Thirring variant, and show that the optimal constants converge to the (fractional) Gagliardo–Nirenberg constants in the strong-coupling limit, with explicit statements on thermodynamic limits. Together, these results extend stability-type bounds to bosonic-like systems with strong local repulsion and connect spectral theory with sharp density-based inequalities for large quantum systems.

Abstract

We prove an analogue of the Lieb--Thirring inequality for many-body quantum systems with the kinetic operator $\sum_i (-Δ_i)^s$ and the interaction potential of the form $\sum_i δ_i^{-2s}$ where $δ_i$ is the nearest-neighbor distance to the point $x_i$. Our result extends the standard Lieb--Thirring inequality for fermions and applies to quantum systems without the anti-symmetry assumption on the wave functions. Additionally, we derive similar results for the Hardy--Lieb--Thirring inequality and obtain the asymptotic behavior of the optimal constants in the strong coupling limit.

Lieb--Thirring inequalities for large quantum systems with inverse nearest-neighbor interactions

TL;DR

This work extends Lieb–Thirring-type energy bounds to many-body quantum systems without antisymmetry by replacing the standard two-body repulsion with an inverse nearest-neighbor interaction and employing a fractional kinetic energy . The authors develop a microlocal localization framework, including local uncertainty and local exclusion principles built on a Besicovitch-based multiscale covering, to derive global density-based bounds. They establish a Lieb–Thirring inequality with the inverse-nearest-neighbor potential, prove a Hardy–Lieb–Thirring variant, and show that the optimal constants converge to the (fractional) Gagliardo–Nirenberg constants in the strong-coupling limit, with explicit statements on thermodynamic limits. Together, these results extend stability-type bounds to bosonic-like systems with strong local repulsion and connect spectral theory with sharp density-based inequalities for large quantum systems.

Abstract

We prove an analogue of the Lieb--Thirring inequality for many-body quantum systems with the kinetic operator and the interaction potential of the form where is the nearest-neighbor distance to the point . Our result extends the standard Lieb--Thirring inequality for fermions and applies to quantum systems without the anti-symmetry assumption on the wave functions. Additionally, we derive similar results for the Hardy--Lieb--Thirring inequality and obtain the asymptotic behavior of the optimal constants in the strong coupling limit.
Paper Structure (8 sections, 11 theorems, 165 equations, 1 figure)

This paper contains 8 sections, 11 theorems, 165 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Global and local Sobolev norms
  3. Construction of covering sub-cubes
  4. Local uncertainty estimates
  5. Local exclusion estimates
  6. Lieb--Thirring inequalities
  7. Hardy--Lieb--Thirring inequalities
  8. Fermionic estimates

Key Result

Theorem 1

Let $d\in \mathbb{N}$, $s >0$ and $\lambda>0$. Then there exists a constant $K_{\rm LT}(d, s, \lambda)>0$ such that for every $N \ge 1$ and every normalized wave function $\Psi\in L^2(\mathbb{R}^{dN})$ we have Moreover, there exist universal constants $C(d,s)>0,\lambda(d,s)$ and $k_1(d,s)>0$ such that the optimal constant in eq:LT-nearest-potential-1-fractional satisfies Here $C_{\rm GN}(d,s)$ i

Figures (1)

  • Figure 1: An enlarged set of a cluster $\Omega_K$ in 2D. The boundary of the enlarged set is mostly flat, except at the corners where it must be smooth. In 2D, there are only two types of conners.

Theorems & Definitions (24)

  • Theorem 1
  • Remark 2: Implication to the fermionic Lieb--Thirring inequality
  • Remark 3: Ground state energy in the thermodynamic limit
  • Theorem 4
  • Definition 5
  • Lemma 6: Extension of Sobolev spaces
  • Lemma 7: Comparison of Sobolev norms
  • proof
  • Lemma 8: Construction of enlarged clusters
  • proof
  • ...and 14 more