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 \(N\rightarrow \infty \).
Dedicated to Herbert Spohn, on the occasion of his 70th birthday.
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Notes
- 1.
Bounding the full kinetic energy from below by that of the one-body density \(\sqrt{\rho _{\Psi _N}}\).
- 2.
There is no blow-up then, but one might want to adapt the method to obtain convergence of density matrices to the stable GP minimizers.
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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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Appendix A. Extension to Anharmonic Potentials
Appendix A. Extension to Anharmonic Potentials
The arguments given in this paper can be extended in various directions. One possibility is to consider anharmonic potentials. For completeness, we state here the corresponding result when the external potential is chosen in the form
but we expect similar results when V has a unique minimizer and behaves like this in a neighborhood of this point, similarly to what was done in [30]. When \(s\ne 2\) the limit \(a\rightarrow a_*\) requires to have \(\Omega =0\). Although a stronger confinement \(s>2\) can control the rotating gas at infinity, it is not sufficient to control rotating effects near the blow-up point. So we do not consider any rotation here. The many-particle Hamiltonian then takes the form
The following can be proved by arguing exactly as we did for \(s=2\).
Theorem 4.4
(Collapse and condensation of the many-body ground state for anharmonic potentials). Let \(\Omega \equiv 0\), \(s>0\), \(c_0>0\), \(0<\beta <1/2\) and \(a_N=a_*-N^{-\alpha }\) with
Let \(\Psi _N\) be the unique ground state of \(\widetilde{H}_N\). Then we have
for all \(k\in \mathbb {N}\), where \(\widetilde{Q}_N\) is the rescaled Gagliardo–Nirenberg optimizer given by
with
In addition, we have
The right side of (A.3) is of course the expansion of the Gross–Pitaevskii energy, which has already been derived in [30].
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Lewin, M., Thành Nam, P., Rougerie, N. (2018). Blow-Up Profile of Rotating 2D Focusing Bose Gases. In: Cadamuro, D., Duell, M., Dybalski, W., Simonella, S. (eds) Macroscopic Limits of Quantum Systems. MaLiQS 2017. Springer Proceedings in Mathematics & Statistics, vol 270. Springer, Cham. https://doi.org/10.1007/978-3-030-01602-9_7
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