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Blow-Up Profile of Rotating 2D Focusing Bose Gases

  • Mathieu LewinEmail author
  • Phan Thành Nam
  • Nicolas Rougerie
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 270)

Abstract

We consider the Gross–Pitaevskii equation describing an attractive Bose gas trapped to a quasi 2D layer by means of a purely harmonic potential, and which rotates at a fixed speed of rotation \(\Omega \). First, we study the behavior of the ground state when the coupling constant approaches \(a_*\), the critical strength of the cubic nonlinearity for the focusing nonlinear Schrödinger equation. We prove that blow-up always happens at the center of the trap, with the blow-up profile given by the Gagliardo–Nirenberg solution. In particular, the blow-up scenario is independent of \(\Omega \), to leading order. This generalizes results obtained by Guo and Seiringer (Lett. Math. Phys., 2014, vol. 104, p. 141–156) in the nonrotating case. In a second part, we consider the many-particle Hamiltonian for N bosons, interacting with a potential rescaled in the mean-field manner \(-a_NN^{2\beta -1}w(N^\beta x)\), with \(w\geqslant 0\) a positive function such that \(\int _{{\mathbb R }^2}w(x)\,dx=1\). Assuming that \(\beta <1/2\) and that \(a_N\rightarrow a_*\) sufficiently slowly, we prove that the many-body system is fully condensed on the Gross–Pitaevskii ground state in the limit \(N\rightarrow \infty \).

Notes

Acknowledgements

It is our pleasure to dedicate this paper to Herbert Spohn, on the occasion of his 70th birthday. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreements MDFT No 725528 and CORFRONMAT No 758620).

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Mathieu Lewin
    • 1
    Email author
  • Phan Thành Nam
    • 2
  • Nicolas Rougerie
    • 3
  1. 1.CNRS and CEREMADEUniversité Paris-Dauphine, PSL Research UniversityParisFrance
  2. 2.Department of MathematicsLudwig Maximilian University of MunichMunichGermany
  3. 3.CNRS, LPMMC (UMR 5493)Université Grenoble-AlpesGrenobleFrance

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