The quantum-mechanical collapse (alias “fall onto the center” of particles attracted by potential ~ -V/r^2, with V > 0) is a well-known issue in the quantum theory. This work demonstrates that, in a rarefied gas of quantum particles attracted by the above-mentioned potential, the repulsive nonlinearity induced by collisions between the particles prevents the collapse, recreating the missing ground state, in the framework of the mean-field approximation [1]. Further, a phase transition in the ground state is found at a critical value of V. The setting may be realized in the 3D gas of dipolar bosons attracted by a central charge, including the case of the binary gas [2]. The addition of the harmonic-oscillator trapping potential gives rise to a tristability, in the case when the Schroedinger equation still does not lead to the collapse. In the 2D setting, the cubic nonlinearity is not strong enough to prevent the collapse; however, the quintic term does it. The analysis was also extended to the 3D anisotropic setting, with the dipoles polarized by an external uniform field [3].
Finally, an exact numerical analysis of the same model in the framework of the multi-body quantum theory [4] demonstrates that, although the complete suppression of the quantum collapse does not occur, the ground state predicted by the mean-field approximation corresponds to a metastable state which emerges in the framework of the multi-body system.
[1] H. Sakaguchi and B. A. Malomed, Suppression of the quantum-mechanical collapse by repulsive interactions in a quantum gas, Phys. Rev. A 83, 013907 (2011).
[2] H. Sakaguchi and B. A. Malomed, Suppression of the quantum collapse in binary bosonic gases, Phys. Rev. A 88, 043638 (2013).
[3] H. Sakaguchi and B. A. Malomed, Suppression of the quantum collapse in an anisotropic gas of dipolar bosons, Phys. Rev. A 84, 033616 (2011).
[4] G. E. Astrakharchik and B. A. Malomed, Quantum versus mean-field collapse in a many-body system, Phys. Rev. A 92, 043632 (2015).