Rigorous proof of dimension-dependent Thomas–Fermi convergence rates for quantum droplets

Establish a rigorous mathematical proof for the dimension-dependent convergence rates of the Thomas–Fermi approximation in the density-locked extended Gross–Pitaevskii equation for quantum droplets in free space (i.e., V_d ≡ 0), by showing that as the particle number N → ∞ in dimensions d ∈ {1,2,3}, the chemical potential error satisfies |μ_g − μ^s| = O(N^{-1/d}) and the L2-norm error satisfies ||φ_g − φ^s||_2 = O(N^{-1/(2d)}), where φ_g and μ_g are the numerical ground-state wave function and chemical potential, and φ^s and μ^s are their Thomas–Fermi counterparts.

Background

The paper studies ground states of quantum droplets in Bose–Bose mixtures using the extended Gross–Pitaevskii equation with Lee–Huang–Yang corrections and introduces a reduced single-component density-locked model. In the strong-coupling regime in free space, the Thomas–Fermi approximation (TFA) neglects kinetic energy and predicts a flat-top (“liquid drop”) ground-state profile.

Through numerical experiments in dimensions d = 1, 2, and 3, the authors compare computed ground states and chemical potentials to their TFA counterparts and report dimension-dependent convergence rates: approximately O(N^{-1/d}) for the chemical potential error and O(N^{-1/(2d)}) for the L2 error in the wave function as N → ∞. They then explicitly note that a rigorous mathematical justification of these rates remains open.