论文标题
奇异平均场状态：对最新结果的简要回顾
Singular mean-field states: A brief review of recent results
论文作者
论文摘要
本文对最近的发现进行了重点回顾，这些发现在某些情况下表明，在中心的密度模式的奇异性的结合状态存在，而国家的总体规范会融合，而状态具有物理上有意义。这种类型的一种模型是基于2D Gross-Pitaevskii方程（GPE），该方程将有吸引力的电位〜1/r^2和由Lee-huang-yang效应（平均场状态周围的量子波动）引起的四分之一的自我抑制非线性结合在一起。 GPE证明了抑制2D量子崩溃的抑制，这是由稳定的基态（GS）的出现驱动的，其密度在r->0。在r-> 0的模式下具有可集成的奇异性〜1/r^{4/3}。该模型的违反直觉的特点是，即使电势的迹象从吸引力到排斥，也存在GS，只要其强度足够小。这种特殊性发现了一个相关的解释。综述中概述的另一个模型包括1D，2D和3D GPE，分别具有隔膜（第七阶），五分之一和立方自我抑制项。这些方程产生稳定的奇异孤子，它代表每个维度D的GS，密度奇数〜1/r^{2/（4-D）。可以将这种状态视为筛查各自的非线性“裸露”的delta功能有吸引力的潜力。
This article provides a focused review of recent findings which demonstrate, in some cases quite counter-intuitively, the existence of bound states with a singularity of the density pattern at the center, while the states are physically meaningful because their total norm converges. One model of this type is based on the 2D Gross-Pitaevskii equation (GPE) which combines the attractive potential ~ 1/r^2 and the quartic self-repulsive nonlinearity, induced by the Lee-Huang-Yang effect (quantum fluctuations around the mean-field state). The GPE demonstrates suppression of the 2D quantum collapse, driven by the attractive potential, and emergence of a stable ground state (GS), whose density features an integrable singularity ~1/r^{4/3} at r --> 0. Modes with embedded angular momentum exist too, and they have their stability regions. A counter-intuitive peculiarity of the model is that the GS exists even if the sign of the potential is reversed from attraction to repulsion, provided that its strength is small enough. This peculiarity finds a relevant explanation. The other model outlined in the review includes 1D, 2D, and 3D GPEs, with the septimal (seventh-order), quintic, and cubic self-repulsive terms, respectively. These equations give rise to stable singular solitons, which represent the GS for each dimension D, with the density singularity ~1/r^{2/(4-D). Such states may be considered as a result of screening of a "bare" delta-functional attractive potential by the respective nonlinearity.
