The mean-field limit of the Lieb-Liniger model

Matthew Rosenzweig

doi:10.3934/dcds.2022006

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2022, Volume 42, Issue 6: 3005-3037. Doi: 10.3934/dcds.2022006

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The mean-field limit of the Lieb-Liniger model

Matthew Rosenzweig^,

MIT, Department of Mathematics, 77 Massachusetts Ave, Cambridge, MA 02139-4307, USA

^*Corresponding author: Matthew Rosenzweig
^*Corresponding author: Matthew Rosenzweig

Received: July 2021

Revised: November 2021

Early access: January 2022

Published: June 2022

The author acknowledges financial support from the University of Texas at Austin and the Simons Collaboration on Wave Turbulence

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Abstract

We consider the well-known Lieb-Liniger (LL) model for $ N $ bosons interacting pairwise on the line via the $ \delta $ potential in the mean-field scaling regime. Assuming suitable asymptotic factorization of the initial wave functions and convergence of the microscopic energy per particle, we show that the time-dependent reduced density matrices of the system converge in trace norm to the pure states given by the solution to the one-dimensional cubic nonlinear Schrödinger equation (NLS) with an explict rate of convergence. In contrast to previous work [3] relying on the formalism of second quantization and coherent states and without an explicit rate, our proof is based on the counting method of Pickl [65,66,67] and Knowles and Pickl [44]. To overcome difficulties stemming from the singularity of the $ \delta $ potential, we introduce a new short-range approximation argument that exploits the Hölder continuity of the $ N $-body wave function in a single particle variable. By further exploiting the $ L^2 $-subcritical well-posedness theory for the 1D cubic NLS, we can prove mean-field convergence when the limiting solution to the NLS has finite mass, but only for a very special class of $ N $-body initial states.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35Q40.

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Table 1. Notation

Symbol	Definition
$ A\lesssim B, \ A\sim B $	There are absolute constants $ C_1, C_2>0 $ such that $ A\leq C_1 B $ or $ C_2B\leq A\leq C_1 B $; subscript $ p $ (e.g. $ A\lesssim_p B $) denotes dependence on parameter $ p $
$ \underline{x}_{k}, \ \underline{x}_{i;i+k} $	$ (x_1, \ldots, x_k), \ (x_i, \ldots, x_{i+k}) $, where $ x_j\in {\mathbb{R}} $ for $ j\in\left\{ {1, \ldots, k} \right\} $ or $ j\in\left\{ {i, \ldots, i+k} \right\} $
$ d \underline{x}_{k}, \ d \underline{x}_{i;i+k} $	$ dx_1 \cdots dx_k, \ dx_i \cdots dx_{i+k} $
$ {\mathbb{N}}, \ {\mathbb{N}}_0 $	natural numbers, natural numbers inclusive of zero
$ \left\langle {\cdot}\|{\cdot} \right\rangle $	$ L^2( {\mathbb{R}}^N) $ inner product with physicist's convention: $\left\langle {f}\|{g}\right\rangle := \int_{ {\mathbb{R}}^N}d \underline{x}_N \overline{f( \underline{x}_N)}g( \underline{x}_N) $
$ \left\langle \cdot , \cdot \right\rangle $	duality pairing
$ \left\langle {{\cdot}} \right\| \ \left\| {{\cdot}} \right\rangle $	Dirac's bra-ket notation: see footnote 2
$ A_{i_1\cdots i_k}^{(k)} $	subscript denotes that the operator on $ L^2( {\mathbb{R}}^N) $ acts in the variables $ (x_{i_1}, \ldots, x_{i_k}) $
$ \varphi^{\otimes k} $	$ k $-fold tensor product of $ \varphi $ with itself realized as $ \varphi^{\otimes k}( \underline{x}_k) = \prod_{i=1}^k \varphi(x_i), \ \underline{x}_k\in {\mathbb{R}}^k $
$ {\rm{Tr}}_{1, \ldots, N} $	trace on $ L^2( {\mathbb{R}}^N) $
$ {\rm{Tr}}_{k+1, \ldots, N} $	partial trace on $ L^2( {\mathbb{R}}^N) $ over $ x_{k+1}, \ldots, x_N $ coordinates
$ {\bf{1}}, \ {\bf{1}}_N $	identity operator on $ L^2( {\mathbb{R}}) $ and on $ L^2( {\mathbb{R}}^N) $
$ H_{N}, \ H_{N, \varepsilon} $	LL Hamiltonian and regularized LL Hamiltonian: see (1.2) and (1.15)
$ p_j, \ q_j $	projectors $ {\bf{1}}^{\otimes j-1}\otimes p \otimes {\bf{1}}^{N-j}, \ {\bf{1}}^{\otimes j-1} \otimes q \otimes {\bf{1}}^{N-j} $: see (3.2)
$ P_k $	projector onto subspace of $ k $ particles not in the state $ \varphi(t) $: see (3.4)
$ \widehat{f}, \ \widehat{f}^{-1} $	operator $ L^2( {\mathbb{R}}^N)\rightarrow L^2( {\mathbb{R}}^N) $ defined by $ \widehat{f} := \sum_{k=0}^N f(k)P_k $, for $ f: {\mathbb{Z}}\rightarrow {\mathbb{C}} $: see (3.6)
$ n_N, m_N $ $ \widehat{n_N}, \widehat{m_N} $	functions $ {\mathbb{Z}}\rightarrow {\mathbb{C}} $ and operators $ L^2( {\mathbb{R}}^N)\rightarrow L^2( {\mathbb{R}}^N) $: see 3.1
$ \mu, \nu $ $ \widehat{\mu}, \widehat{\nu} $	functions $ {\mathbb{Z}}\rightarrow {\mathbb{C}} $ and operators $ L^2( {\mathbb{R}}^N)\rightarrow L^2( {\mathbb{R}}^N) $: see (3.33) and (3.52)
$ \beta_N $	time-dependent functional of solution $ \varphi $ to (1.8) and $ \Phi_N $ to (1.3): see 3.1
$ \tau_n $	shift operator on $ {\mathbb{C}}^{ {\mathbb{Z}}} $: see (3.16)
$ \underline{\Delta}_k $	Laplacian on $ {\mathbb{R}}^k $: $ \underline{\Delta}_k := \sum_{i=1}^k \Delta_i $
$ \left[ \cdot , \cdot \right]$	commutator bracket: $\left[ A, B \right] := AB-BA $

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