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

Andrew Rout, Vedran Sohinger

Abstract

In this paper, we give a microscopic derivation of Gibbs measures for the focusing cubic nonlinear Schrödinger equation on the one-dimensional torus from many-body quantum Gibbs states. Since we are not making any positivity assumptions on the interaction, it is necessary to introduce a truncation of the mass in the classical setting and of the rescaled particle number in the quantum setting. Our methods are based on a perturbative expansion of the interaction, similarly as in previous work of Fröhlich, Knowles, Schlein, and the second author. Due to the presence of the truncation, the obtained series have infinite radius of convergence. We treat the case of bounded, integrable, and delta function interaction potentials, without any sign assumptions. Within this framework, we also study time-dependent correlation functions. This is the first such known result in the focusing regime.

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

Abstract

In this paper, we give a microscopic derivation of Gibbs measures for the focusing cubic nonlinear Schrödinger equation on the one-dimensional torus from many-body quantum Gibbs states. Since we are not making any positivity assumptions on the interaction, it is necessary to introduce a truncation of the mass in the classical setting and of the rescaled particle number in the quantum setting. Our methods are based on a perturbative expansion of the interaction, similarly as in previous work of Fröhlich, Knowles, Schlein, and the second author. Due to the presence of the truncation, the obtained series have infinite radius of convergence. We treat the case of bounded, integrable, and delta function interaction potentials, without any sign assumptions. Within this framework, we also study time-dependent correlation functions. This is the first such known result in the focusing regime.
Paper Structure (36 sections, 40 theorems, 265 equations)

This paper contains 36 sections, 40 theorems, 265 equations.

Table of Contents

  1. Introduction
  2. Setup
  3. The Classical Problem
  4. The Quantum Problem
  5. Statement of the results
  6. The time-independent problem
  7. The time-dependent problem
  8. Previously known results
  9. Main ideas of the proofs
  10. Organisation of the paper
  11. Notation and auxiliary results
  12. Notation
  13. Auxiliary Results
  14. The time-independent problem with bounded interaction potential. Proof of Theorem \ref{['bounded_trace_class_final_convergence_theorem']}.
  15. Basic Estimates
  16. ...and 21 more sections

Key Result

Proposition 1.2

Let $\varphi$ be as in random_classical_initial. Given $g \in H^{-\frac{1}{2} + \varepsilon}$ for $\varepsilon>0$, we let $\varphi(g) := \langle g, \varphi \rangle$ and $\overline{\varphi}(g) := \langle \varphi,g \rangle$. Furthermore, we let $(\varphi)^*(g)$ denote either $\varphi(g)$ or $\overline where the sum is taken over all complete pairings of $\{1,\ldots,n\}$, and where edges of $\Pi$ are

Theorems & Definitions (77)

  • Proposition 1.2
  • Theorem 1.4: Convergence for $w \in L^{\infty}(\mathbb{T})$
  • Theorem 1.5: Convergence for $w \in L^1(\mathbb{T})$
  • Theorem 1.6: Convergence for $w=-\delta$
  • Remark 1.7
  • Definition 1.8
  • Definition 1.9
  • Theorem 1.10: Convergence for $w \in L^{\infty}(\mathbb{T})$
  • Theorem 1.11: Convergence for $w \in L^1(\mathbb{T})$
  • Theorem 1.12: Convergence for $w=-\delta$
  • ...and 67 more