AI 中文总结
本文研究奇异势下的量子态，揭示吸引型奇异反比平方势在三维玻色气体中可抑制量子坍缩并生成基态，还发现增长快于负简谐势的排斥势会产生全套局域束缚态，为相关研究提供展望。
AI 中文摘要
已知在三维（3D）线性薛定谔方程框架下，吸引型奇异反比平方势会引发临界量子坍缩。本文总结的理论结果表明，在携带电偶极矩、被中心电荷吸引且粒子间存在排斥接触相互作用的三维玻色气体中，该奇异势导致的坍缩会被抑制，且会生成原本缺失的基态（GS）。在平均场近似下，排斥相互作用由对应的格罗斯-皮塔耶夫斯基（GP）方程中的立方项表示。除基态外，本文还简要考虑了具有角动量的激发态。文章探讨的另一主题是r→∞时存在奇异性的排斥势下，线性薛定谔方程与GP方程中的一维（1D）和二维（2D）束缚态。一项最新结果显示，这种增长快于负简谐势的排斥势，会产生一整套违反直觉的可归一化（局域）束缚态。文章提出了对在r→0或r→∞处具有奇异性的势作用下存在的线性与非线性束缚态开展进一步研究的展望。
英文摘要
It is known that the attractive singular inverse-square potential gives rise to the critical quantum collapse in the framework of the three-dimensional (3D) linear Schroedinger equation. This article summarizes theoretical results which demonstrate suppression of the collapse, caused by this singular potential, and the creation of the otherwise missing ground state (GS) in a 3D gas of bosonic particles, carrying an electric dipole moment, which are pulled to the central electric charge, with repulsive contact interactions between the particles. In the mean-field approximation, the repulsive interactions are represented by the cubic term in the respective Gross-Pitaevskii (GP) equation. In addition to the GS, excited states with angular momentum are briefly considered too. Another topic considered in the article is 1D and 2D bound states in the linear Schroedinger and GP equations with the repulsive potential, which demonstrates a singularity at r --> infinity. A very recent result is that such a potential, growing faster than the negative harmonic-oscillator potential, produces a full spectrum of counter-intuitive normalizable (localized) bound states. The article puts forward perspectives for further studies of linear and nonlinear bound states existing under the action of the potentials with the singularity at r --> 0 or r --> infinity.
Commentsto be published as a perspective article in Advanced Physics Research
