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On the Schrödinger–Lohe Hierarchy for Aggregation and Its Emergent Dynamics

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Abstract

The Lohe hierarchy is a hierarchy of finite-dimensional aggregation models consisting of the Kuramoto model, the complex Lohe sphere model, the Lohe matrix model and the Lohe tensor model. In contrast, the Schrödinger–Lohe model is the only known infinite-dimensional Lohe aggregation model in literature. In this paper, we provide an explicit connection between the Schrödinger–Lohe model and the complex Lohe sphere model, and then by exploiting this explicit relation, we construct infinite-dimensional liftings of the Lohe matrix and the Lohe tensor models. In this way, we establish the Schrödinger–Lohe hierarchy which corresponds to the infinite-dimensional extensions of the Lohe hierarchy. For the proposed hierarchy, we provide sufficient frameworks leading to the complete aggregation in terms of coupling strengths and initial configurations.

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Acknowledgements

The work of S.-Y.Ha is supported by NRF-2020R1A2C3A01003881. The work of H. Park is supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2019R1I1A1A01059585)

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Authors and Affiliations

  1. Department of Mathematical Science and Research Institute of Mathematics, Seoul National University, Seoul, 08826, Republic of Korea

    Seung-Yeal Ha

  2. Korea Institute for Advanced Study, Hoegiro 85, Seoul, 02455, Republic of Korea

    Seung-Yeal Ha

  3. Department of Mathematical Sciences, Seoul National University, Seoul, 08826, Republic of Korea

    Hansol Park

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  1. Seung-Yeal Ha
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  2. Hansol Park
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Correspondence to Hansol Park.

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Communicated by Eric A. Carlen.

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Ha, SY., Park, H. On the Schrödinger–Lohe Hierarchy for Aggregation and Its Emergent Dynamics. J Stat Phys 181, 2150–2190 (2020). https://doi.org/10.1007/s10955-020-02659-0

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