Blow-Up Profile of Rotating 2D Focusing Bose Gases

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.

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

$$V(x)=c_0|x|^s$$

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

$$\begin{aligned} \widetilde{H}_N = \sum _{j=1} ^N \left( -\Delta _{x_j} + c_0|x_j|^s\right) - \frac{a}{N-1} \sum _{1\leqslant i<j \leqslant N}w_N(x_i-x_j). \end{aligned}$$

(A.1)

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

$$0<\alpha < \min \left\{ \frac{ \beta (s+2)}{s+3}, \frac{(1-2\beta )(s+2)}{s}\right\} .$$

Let $\Psi _N$ be the unique ground state of $\widetilde{H}_N$. Then we have

$$\begin{aligned} \lim _{N \rightarrow \infty } {{\mathrm{\mathrm{Tr}}}}\Big | \gamma _{\Psi _{N} }^{(k)} - |\widetilde{Q}_N^{\otimes k} \rangle \langle \widetilde{Q}_{N}^{\otimes k}| \Big |=0, \end{aligned}$$

(A.2)

for all $k\in \mathbb {N}$, where $\widetilde{Q}_N$ is the rescaled Gagliardo–Nirenberg optimizer given by

$$\widetilde{Q}_N(x)= (a_*)^{-1/2}\widetilde{\lambda }(a_*-a_N)^{-\frac{1}{2+s}} Q\left( \widetilde{\lambda }(a_*-a_N)^{-\frac{1}{2+s}} x\right) ,$$

with

$$\widetilde{\lambda }=\left( \frac{s}{2} c_0\int _{{\mathbb R }^2} |x|^s |Q(x)|^2 dx \right) ^{\frac{1}{2+s}}.$$

In addition, we have

$$\begin{aligned} \frac{\min \sigma (\widetilde{H}_N)}{N}= (a_*-a_N)^{\frac{s}{s+2}} \Big ( \frac{\widetilde{\lambda }^2}{a_*}\frac{s+2}{s} + o(1)\Big ). \end{aligned}$$

(A.3)

The right side of (A.3) is of course the expansion of the Gross–Pitaevskii energy, which has already been derived in [30].