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The mean-field limit of the Lieb-Liniger model

  • *Corresponding author: Matthew Rosenzweig

    *Corresponding author: Matthew Rosenzweig

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

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  • 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.

    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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