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Gibbs Measures of Nonlinear Schrödinger Equations as Limits of Many-Body Quantum States in Dimensions \({d \leqslant 3}\)

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Abstract

We prove that Gibbs measures of nonlinear Schrödinger equations arise as high-temperature limits of thermal states in many-body quantum mechanics. Our results hold for defocusing interactions in dimensions \({d =1,2,3}\). The many-body quantum thermal states that we consider are the grand canonical ensemble for d = 1 and an appropriate modification of the grand canonical ensemble for \({d =2,3}\). In dimensions d = 2, 3, the Gibbs measures are supported on singular distributions, and a renormalization of the chemical potential is necessary. On the many-body quantum side, the need for renormalization is manifested by a rapid growth of the number of particles. We relate the original many-body quantum problem to a renormalized version obtained by solving a counterterm problem. Our proof is based on ideas from field theory, using a perturbative expansion in the interaction, organized by using a diagrammatic representation, and on Borel resummation of the resulting series.

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Authors and Affiliations

  1. Institute for Theoretical Physics, ETH Zürich, Zurich, Switzerland

    Jürg Fröhlich

  2. Department of Mathematics, University of Geneva, Geneva, Switzerland

    Antti Knowles

  3. Department of Mathematics, University of Zürich, Zurich, Switzerland

    Benjamin Schlein

  4. Mathematics Institute, University of Warwick, Coventry, UK

    Vedran Sohinger

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  1. Jürg Fröhlich
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  2. Antti Knowles
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Correspondence to Benjamin Schlein.

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Fröhlich, J., Knowles, A., Schlein, B. et al. Gibbs Measures of Nonlinear Schrödinger Equations as Limits of Many-Body Quantum States in Dimensions \({d \leqslant 3}\) . Commun. Math. Phys. 356, 883–980 (2017). https://doi.org/10.1007/s00220-017-2994-7

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