The talk aims to produce an overview of results for states with an integrable singularity of the density at $r\rightarrow 0$, trapped in the potential of attraction of an electric dipole to a central charge, $% U(r)=-(U_{0}/2)r^{-2}$. The linear Schrodinger equation with this potential gives rise to the quantum collapse at $U_{0}>1/4$ in 3D, and at any $U_{0}>0$ in 2D. On the contrary, the 3D Gross-Pitaevskii equation (GPE) with the cubic self-repulsion suppresses the collapse and creates a ground state (GS) with an integrable (in 3D) density singularity $% \sim r^{-2}$, at all values of $U_{0}>0$ [1]. In the framework of the quantum many-body theory, this mean-field (MF) GS corresponds to a metastable state secured against the collapse by a high potential barrier [2]. In 2D, the GS, with integrable density singularity $\sim r^{-4/3}$, is created by the Lee-Huang-Yang-Petrov quartic self-repulsive term, which represents a correction to the MF produced by quantum fluctuations [3]. The same 2D setting supports singular vortex states with orbital quantum number $l$. These states are stable if the strength of the attractive potential is large enough, viz., $U_{0}>(7/9)(3l^{2}-1)$ . Similarly, "antidark" stable vortex states, with the same density singularity, $\sim r^{-4/3}$, at $r\rightarrow 0$, have been constructed on top of a modulationally stable background with a nonzero density at $r\rightarrow \infty $ [4].
References
[1] H. Sakaguchi and B. A. Malomed, Suppression of the quantum-mechanical collapse by repulsive interactions in a quantum gas, Phys. Rev. A \textbf{83}% , 013607 (2011).
[2] G. E. Astrakharchik and B. A. Malomed, Quantum versus mean-field collapse in a many-body system, Phys. Rev. A \textbf{92}, 043632 (2015).
[3] E. Shamriz, Z. Chen, and B. A. Malomed, Suppression of the quasi-two-dimensional quantum collapse in the attraction field by the Lee-Huang-Yang effect, Phys. Rev. A \textbf{101}, 063628 (2020).
[4] Z. Chen and B. A. Malomed, Singular and regular vortices on top of a background pulled to the center, J. Optics \textbf{23}, 074001 (2021).