Dynamical phase retrieval for Schr{\"o}dinger evolution on finite graphs

We study dynamical phase retrieval for Schr\''odinger evolutions on finite connected graphs.

Let \[ H\_Q=\Delta\_G+Q \] 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 $\lambda\_j+\lambda\_k$ determine the unordered pair $\{j,k\}$, invertibility of the squared-eigenvector matrix $\bigl(\phi\_k(j)^2\bigr)\_{j,k}$ 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.