AI 中文总结
研究三维聚焦能量临界薛定谔方程圆柱对称类的低于阈值散射，通过分离孤子状紧致临界元素的两个刚性入口，利用位力估计等排除入口，经紧致性约化得到条件圆柱阈值散射准则。
AI 中文摘要
研究了三维聚焦能量临界薛定谔方程在圆柱对称类中的低于阈值散射。对于孤子状紧致临界元素，分离出两个相关的刚性入口：轴向中心的最佳固定轴窗口条件和低频尾部小条件。通过移动的局部位力估计和轴向零动量归一化后的有限质量位力论证排除这些入口，通过紧致性约化得到一个条件圆柱阈值散射准则。
英文摘要
Below-threshold scattering is studied for the three-dimensional focusing energy-critical Schrödinger equation in the cylindrically symmetric class. For soliton-like compact critical elements, two linked rigidity entrances are isolated: a best fixed-axis window condition for the axial center and a low-frequency tail smallness condition. A shifted localized virial estimate and a finite-mass virial argument after axial zero-momentum normalization exclude these entrances, yielding a conditional cylindrical threshold scattering criterion via compactness reduction.
