A microscopic derivation of Gibbs measures for the 1D focusing quintic nonlinear Schrödinger equation

TL;DR

The authors develop a rigorous bridge between quantum many-body Gibbs states with a three-body interaction and the Gibbs measures for the focusing quintic NLS on the 1D torus. They implement a mass/particle-number truncation and employ a perturbative expansion, Feynman-Kac bounds, and diagrammatic techniques to prove both time-independent and time-dependent convergence results in bounded, L^1, and delta interaction regimes. A key novelty is the first microscopic derivation of time-dependent correlation functions for quintic NLS, including unbounded interactions via controlled approximations, tying the quantum and classical Gibbs descriptions together. The work extends the established cubic/quartic analyses to the quintic regime, providing a foundational framework for almost-sure global well-posedness via Gibbs invariance in the focusing setting and offering tools for future exploration of higher-order nonlinearities in one dimension and beyond.

Abstract

In this work, we obtain a microscopic derivation of Gibbs measures for the focusing quintic nonlinear Schrödinger equation (NLS) on $\mathbb{T}$ from many-body quantum Gibbs states. On the quantum many-body level, the quintic nonlinearity corresponds to a three-body interaction. This is a continuation of our previous work. In the aforementioned work, we studied the cubic problem, which corresponds to a two-body interaction on the quantum many-body level. In our setup, we truncate the mass of the classical free field in the classical setting and the rescaled particle number in the quantum setting. Our methods are based on a perturbative expansion previously developed in the work of Fröhlich, Knowles, Schlein, and the second author. We prove results both in the time-independent and time-dependent setting. This is the first such known result in the three-body regime. Furthermore, this gives the first microscopic derivation of time-dependent correlation functions for Gibbs measures corresponding to the quintic NLS, as studied in the work of Bourgain.

Figures (4)

• Figure 1: An unpaired graph from Definition \ref{['q_vertex_set_defn']} with $m=p=3$. The black dots correspond to factors of $\varphi_\tau(\cdot)$ and $\varphi_\tau^*(\cdot)$. The wavy lines correspond to factors of the interaction potential $w$.
• Figure 2: A graph with $m=p=3$ with a valid pairing from Definition \ref{['q_pairing_defn']}.
• Figure 3: An example of the coloured graph from Definition \ref{['q_colored_graph_defn']} with the same pairing as Figure \ref{['Fig:uncollapsed_paired']}. The wider dotted edges have colour $-1$, and the finer dotted lines have colour $1$.
• Figure 4: An unpaired graph with $m=2$, $p=3$ corresponding to Definition \ref{['q_delta_graph_defn']}.

Theorems & Definitions (129)

• Lemma 1.3
• proof
• Remark 1.4
• Proposition 1.5: Wick's theorem
• Remark 1.7
• Theorem 1.8: Convergence for bounded interaction potentials
• Theorem 1.9: Convergence for $L^1$ interaction potentials
• Theorem 1.10: Convergence for delta function interaction potentials
• Remark 1.11
• Theorem 1.12: Convergence for bounded potentials