### Abstract

The quantum-mechanical collapse (alias fall onto the center of particles attracted by potential -r^{-2}) is a well-known issue in quantum theory. It is closely related to the quantum anomaly, i.e., breaking of the scaling invariance of the respective Hamiltonian by quantization. We demonstrate that the mean-field repulsive nonlinearity prevents the collapse and thus puts forward a solution to the quantum-anomaly problem that differs from that previously developed in the framework of the linear quantum-field theory. This solution may be realized in the 3D or 2D gas of dipolar bosons attracted by a central charge and in the 2D gas of magnetic dipoles attracted by a current filament. In the 3D setting, the dipole-dipole interactions are also taken into regard, in the mean-field approximation, resulting in a redefinition of the scattering length which accounts for the contact repulsion between the bosons. In lieu of the collapse, the cubic nonlinearity creates a 3D ground state (GS), which does not exist in the respective linear Schrödinger equation. The addition of the harmonic trap gives rise to a tristability, in the case when the Schrödinger 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, creating the GS, as well as its counterparts carrying the angular momentum (vorticity). Counterintuitively, such self-trapped 2D modes exist even in the case of a weakly repulsive potential r^{-2}. The 2D vortical modes avoid the phase singularity at the pivot (r=0) by having the amplitude diverging at r→0 instead of the usual situation with the amplitude of the vortical mode vanishing at r→0 (the norm of the mode converges despite of the singularity of the amplitude at r→0). In the presence of the harmonic trap, the 2D quintic model with a weakly repulsive central potential r^{-2} gives rise to three confined modes, the middle one being unstable, spontaneously developing into a breather. In both the 3D and 2D cases, the GS wave functions are found in a numerical form and in the form of an analytical approximation, which is asymptotically exact in the limit of the large norm.

Original language | English |
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Article number | 013607 |

Journal | Physical Review A - Atomic, Molecular, and Optical Physics |

Volume | 83 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 14 2011 |

### All Science Journal Classification (ASJC) codes

- Atomic and Molecular Physics, and Optics