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Dynamical Hartree–Fock–Bogoliubov Approximation of Interacting Bosons

Abstract

We consider a many-body Bosonic system with pairwise particle interaction given by \(N^{3\beta -1}v(N^\beta x)\) where \(0<\beta <1\) and v a non-negative spherically symmetric function. Our main result is the extension of the local-in-time Fock space approximation of the exact dynamics of squeezed states proved in Grillakis and Machedon (Commun Partial Differ Equ 42(1):24–67, 2017) for \(0<\beta <\frac{2}{3}\) to a global-in-time approximation for \(0<\beta <1\). Our work can also be viewed as a generalization of the results in Boccato et al. (Ann Henri Poincaré 18(1):113–191, 2017) to a more general set of initial data that includes coherent states along with an improved error estimate. The key ingredients in establishing the Fock space approximation are the work of Grillakis and Machedon on the the local well-posedness theory (Grillakis and Machedon in Commun Partial Differ Equ 44(12):1431–1465, 2019), some recent established global estimate in Chong et al. (Commun Partial Differ Equ 56:1–41, 2021), and our quantitative results on the uniform in N global well-posedness of the time-dependent Hartree–Fock–Bogoliubov system.

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Notes

  1. 1.

    We work under the assumption that \(\hbar =1\) and \(2m=1\), where m is the mass. However, it would be interesting to incorporate \(\hbar \) in the calculation to see the explicit dependence of \(\hbar \) in our results. Moreover, as written, (1.1) models a system of interacting particles in the mean-field scaling regime. Cf. [29, Section 1.8].

  2. 2.

    We adopt the standard notation \(A \lesssim B\) to mean there exists a constant, depending on some parameters, such that \(A \le CB\).

  3. 3.

    One should note that the main result in Rodnianski and Schlein’s paper is their result on the rate of convergence of the one-particle Fock space reduced density operator toward the Hartree dynamics. Whereas, the significance of Kuz’s paper is that she was able to show that the mean-field estimate is actually valid for a much longer period of time then most proceeding results had indicated.

  4. 4.

    Cf. [46, Chapter 10] for the definition of quasifree.

  5. 5.

    It should be warned that we follow the convention of [23] and define our annihilation operator to be a linear map as opposed to the conventional definition of anti-linear. This definition is also consistent with the view that \(a_x\) is a distribution-valued operator since \(a_x\psi \) acts linearly on \({\mathcal {F}}\).

  6. 6.

    In the mathematical physics literature, \(e^{{\mathcal {B}}}\) is called the infinite-dimensional Segal–Shale–Weil representation of the double cover of the group of symplectic matrices of integral operators. The elements of the corresponding \(C^*\)-algebra are called Bogoliubov transformations (cf. [21, Chapter 4] and [13, Chapter 11]).

  7. 7.

    In general, one could define the trace density \(\varrho _F(x)=\sum _{j}\lambda _j|\phi _j(x)|^2\) for a self-adjoint trace class (integral) operator F by considering the eigenfunction expansion \(F=\sum _j \lambda _j |\phi _j \rangle \langle \phi _j |\) and noticing that \({\text {Tr}}\left( F\right) = \int _{{\mathbb {R}}^3}\mathrm {d}x\, \{\sum _{j}\lambda _j|\phi _j(x)|^2\}= \sum _j \lambda _j\). However, the connection between \(\varrho _F(x)\) and F(xx) is not immediately obvious when F(xy) is rough. Nevertheless, it has been shown that if \(F = J\circ K\) is trace class and JK are Hilbert–Schmidt operators, then \(\varrho _F(x) = (J\circ K)(x, x)\) a.e.. (Cf. [7, Theorem 3.3]).

    In the physics literature, \(N\varrho _\Gamma (x)\) is called the total-number density and is often denoted by n(x). Here we adopt the standard notation, \(n_{\mathrm {c}}\) and \({\widetilde{n}}\) denote the condensate density and the non-condensate density, respectively, i.e., \(n(x) = n_{\mathrm {c}}(x)+{\widetilde{n}}(x)\).

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Acknowledgements

We would like to thank Prof. Manoussos Grillakis and Prof. Matei Machedon for the beneficial communications and encouragement. Moreover, we highly appreciate the referees for their patience and careful review of the manuscript. Their suggestions have greatly improved the overall quality of our exposition. Some of the work was done while the second author was moving from the University of Maryland to the Beijing Institute of Technology, so he appreciates the kind supports of both institutes. J. Chong was supported by the NSF through the RTG Grant DMS-RTG 1840314. Z. Zhao was supported by UMD’s postdoc support, the Beijing Institute of Technology Research Fund Program for Young Scholars and NSFC-12101046.

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Chong, J.J., Zhao, Z. Dynamical Hartree–Fock–Bogoliubov Approximation of Interacting Bosons. Ann. Henri Poincaré (2021). https://doi.org/10.1007/s00023-021-01100-w

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