Speaker
Prof.
Boris Malomed
(Tel Aviv University, Faculty of Engineering, Dept. of Physical Electronics)
Description
The quantum-mechanical collapse (alias "fall onto the
center" of particles attracted by potential -1/r^2) is a well-
known issue in the elementary quantum theory. It is
closely related to the so-called "quantum anomaly", i.e.,
breaking of the scaling invariance of the respective
Hamiltonian by the quantization. We demonstrate that, in a
rarefied gas of quantum particles attracted by the above-
mentioned potential, the mean-field repulsive nonlinearity
induced by collisions between the particles prevents the
collapse, and thus puts forward a solution to the quantum-
anomaly problem different 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 also in the 2D gas
of magnetic dipoles attracted by a current filament. In lieu
of the collapse, the cubic nonlinearity creates a 3D ground
state (GS), which does not exist in the respective linear
Schroedinger equation. The addition of the harmonic
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 is also extended to the 3D anisotropic setting,
with the dipoles polarized by an external uniform field.
Publications:
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);
H. Sakaguchi and B. A. Malomed, Suppression of the
quantum collapse in an anisotropic gas of dipolar bosons,
Phys. Rev. A 84, 033616 (2011).
Primary author
Prof.
Boris Malomed
(Tel Aviv University, Faculty of Engineering, Dept. of Physical Electronics)
