AI 中文总结
该研究针对有限连通图上的薛定谔演化，基于算子特征值与特征向量给出无相位数据确定初始态的唯一性判据，证明其一般性并给出阻碍唯一性的情形。
AI 中文摘要
我们研究有限连通图上薛定谔演化的动力学相位检索问题。设$H_Q=\triangle_G+Q$为带有实对角势的图薛定谔算子，我们探究由相关薛定谔演化$|e^{-itH_Q}u_0(j)|$（$0\leq t\leq T$，$j\in V$）得到的无相位数据何时能确定初始态$u_0\in\mathbb{C}^V$（仅差一个全局相位）。我们基于$H_Q$的特征值和特征向量给出一个唯一性判据：该判据假设谱满足$B_2$条件（即和式$\lambda_j+\lambda_k$确定无序对$\{j,k\}$）、平方特征向量矩阵$(\phi_k(j)^2)_{j,k}$可逆，以及特征向量对的支集满足重叠条件。在这些假设下，无相位薛定谔数据可唯一确定每个初始态（仅差全局相位）。随后我们证明该判据既存在实例又具有一般性：每个有限连通图都存在显式实对角势使判据成立，且对每个有限连通图，动力学相位检索对勒贝格测度下几乎所有实势$Q\in\mathbb{R}^V$及任意$T>0$均成立，同时我们也给出了若干唯一性的阻碍情形。
英文摘要
We study dynamical phase retrieval for Schr\''odinger evolutions on finite connected graphs. Let \[ H\_Q=Δ\_G+Q \] be a graph Schr\''odinger operator with a real diagonal potential. We investigate when phaseless data obtained from the associated Schr\''odinger evolution \[ |e^{-itH\_Q}u\_0(j)|, \qquad 0\leq t\leq T,\ j\in V, \] determines the initial state $u\_0\in\C^V$ up to a global phase. We give a uniqueness criterion in terms of the eigenvalues and eigenvectors of $H\_Q$. The assumptions are a $B\_2$ condition on the spectrum, meaning that the sums $λ\_j+λ\_k$ determine the unordered pair $\{j,k\}$, invertibility of the squared-eigenvector matrix $\bigl(ϕ\_k(j)^2\bigr)\_{j,k}$ and an overlap condition on the supports of pairs of eigenvectors. Under these hypotheses, the phaseless Schr\''odinger data determine every initial state uniquely, modulo global phase. We then show that the criterion is both realized and generic. Every finite connected graph admits an explicit real diagonal potential for which the criterion holds. Moreover, for every finite connected graph, dynamical phase retrieval holds for Lebesgue-almost every real potential $Q\in\R^V$ and every $T>0$. We also give several obstructions to uniqueness.
CommentsA.M. was supported in part by the National Science Foundation grant DMS-2453769, and an AMS-Simons Research Enhancement Grant
