AI 中文总结
研究分数阶薛定谔方程的逆障碍问题，利用方程非局部特性解决自由边界难题，通过障碍测量相等推出\((q_1 - q_2)u = 0\)，进而恢复势，还证明几何覆盖定理表明可数测量可全局确定势。
AI 中文摘要
我们研究分数阶薛定谔算子\((-\Delta)^s + q\)（\(0 < s < 1\)）的逆障碍问题。对于每个外部数据，状态在有界域中受规定障碍约束，仅在相关非接触集中满足分数阶薛定谔方程。该集合未知且依赖于系数，所以外部狄利克雷 - 诺伊曼映射是非线性的。我们表明方程的非局部特性为解决自由边界移动难题提供了直接方法。外部开集上一个障碍测量相等迫使整个空间中两个相应障碍状态相等，在其公共非接触集中可得\((q_1 - q_2)u = 0\)。当\(s\in[1/4,1)\)时，通过可测唯一延拓可在暴露区域恢复势；当势连续时，结合通常唯一延拓原理和连续性恢复势。我们还证明了非负势的几何覆盖定理：一个非平凡非负外部数据的有理正缩放可暴露整个域直至零测集。因此，在相应假设下，可数个非线性外部障碍测量可全局确定势。
英文摘要
We study an inverse obstacle problem for the fractional Schrödinger operator $(-Δ)^s+q$, $0<s<1$. For each exterior datum, the state is constrained by a prescribed obstacle in a bounded domain and satisfies the fractional Schrödinger equation only in the associated noncontact set. This set is unknown and depends on the coefficient, so the exterior Dirichlet-to-Neumann map is nonlinear. We show that the nonlocal character of the equation gives a direct way around this moving-free-boundary difficulty. Equality of one obstacle measurement on an exterior open set forces equality of the two corresponding obstacle states in the whole space. In their common noncontact set one then obtains $(q_1-q_2)u=0$. This identity yields recovery of the potential in the exposed region by measurable unique continuation when $s\in[1/4,1)$, and by the usual unique continuation principle together with continuity when the potentials are continuous. We also prove a geometric coverage theorem for nonnegative potentials: rational positive scalings of one nontrivial nonnegative exterior datum expose the whole domain up to a null set. Consequently, under the corresponding assumptions, a countable family of nonlinear exterior obstacle measurements determines the potential globally.
