走向现代数学学术报告 – 程汗博士（No. 985)

报告题目：Endpoint Mapping Properties of Wave Operators for Two-Dimensional Schrödinger Operators

报 告 人：程汗 博士（北京应用物理与计算数学研究所）

邀 请 人：王大斌 教授

报告时间：2026年10月28日 14:30

报告地点：东海岸校区D209

报告摘要：While the endpoint theory of Schrödinger wave operators is largely established in other dimensions, their boundedness at p=1 and p=∞ in dimension two has remained a longstanding open problem. We establish sharp endpoint mapping properties for the wave operators W_±(H,-Δ) of two-dimensional Schrödinger operators H=-Δ+V. Together with the known non-endpoint theory, our results give a complete L^p classification under the stated decay assumptions. For real-valued potentials satisfying |V(x)|≲⟨x⟩^-6-, the regular case admits neither strong endpoint bound, but satisfies the substitute estimates

L¹⇒L^1,∞, H¹⇒L¹, L^∞⇒BMO.

In contrast, a first-kind threshold singularity yields boundedness on L^p for every 1≤ p≤∞. Thus an s-wave resonance improves the endpoint behavior in dimension two, in contrast with the favorable role of the regular case in higher dimensions. This reversal supports the viewpoint that endpoint boundedness is favored when H and H₀ have matching spectral structures at zero: the two-dimensional free operator itself has an s-wave resonance, given by the constant function.

Under |V(x)|≲⟨x⟩^-8-, we also determine the endpoint behavior in the remaining cases. A p-wave resonance obstructs both strong endpoint bounds. In the zero-eigenvalue case, L¹ boundedness holds if and only if an s-wave resonance is present and no p-wave resonance occurs; L^∞ boundedness requires, in addition, the vanishing of the degree-two harmonic moments of Vψ for every zero-energy eigenfunction ψ.

报告人简介： 程汗，应用数学博士。本科毕业于华中科技大学，2016—2024 年在华中科技大学完成硕博连读并获博士学位（导师郑权教授），2024年至2026年在北京应用物理与计算数学研究所从事博士后研究（合作导师苗长兴教授、郑继强研究员）。研究方向为调和分析、偏微分方程与数学物理，主要研究薛定谔方程与狄拉克方程的色散估计、波算子理论。在 Mathematische Annalen、Journal of Functional Analysis、Advances in Mathematics等期刊发表论文多篇。