math.AP — Analysis of PDEs

Phase retrieval for stationary Schrödinger evolutions

Contributed by Ben Pineau, João Pedro Ramos, Mitchell A. Taylor

We prove three phase-retrieval results for stationary one-dimensional Schr\"odinger evolutions. Masuda's unique-continuation theorem yields phase retrieval for finite-energy solutions and a broad class of semibounded potentials which may grow at infinity. For real potentials in the Faddeev class $L^1_1(\mathbb R)$, measurements on all of spacetime determine arbitrary $L^2$ initial data. If the potential is smooth and compactly supported to the left of a point, measurements on the exterior half-line to its right suffice.