AI 中文总结
本文证明d≥5的大初值聚焦H^{1/2}-临界非线性薛定谔方程的解在有限多任意小时空锥外散射，给出两个关键证明要素及提升机制，为后续研究提供工具。
AI 中文摘要
本文证明，在维数d≥5的聚焦H^{1/2}-临界非线性薛定谔方程的解，在有限多个任意小的时空锥外发生散射。我们将讨论定理\ref{thm:finite-bad-directions}与孤子分解之间的关系。证明中有两个关键要素：定理\ref{thm:cone-scattering}中的局域化低于基态散射，以及定理\ref{thm:measure-convergence}中对残差部分（即减去散射部分后的部分）渐近行为的刻画。作为副产品，我们还建立了一个提升机制：若解在某时空锥内的一个时间序列上低于基态（或小），则在缩小后的时空锥内，其在大时间上也一致低于基态（或小）。我们预计这将对未来使用集中紧性和时空锥方法的研究有用。
英文摘要
In this article we prove that the solution to the focusing $H^{1/2}$-critical nonlinear Schrödinger equation in dimension $d\geq 5$ scatters outside finitely many arbitrarily small spacetime cones. We will discuss the relationship between Theorem~\ref{thm:finite-bad-directions} and the resolution of solitons. There are two key ingredients in the proof: the localized below-ground-state scattering in Theorem~\ref{thm:cone-scattering}, and the characterization in Theorem~\ref{thm:measure-convergence} of the asymptotic behaviour of the residual part (i.e., after subtracting the scattering part). As a byproduct, we also establish an upgrading machinery: if the solution is below the ground state (resp. small) along a time sequence in a spacetime cone, then it is also below ground state (resp. small) uniformly for large time in a shrinked spacetime cone. We expect this to be useful in future works using concentration compactness and our spacetime cone approach.
