On recovering of solutions of Schr\"odinger equations from their time averages

Abstract

The paper study a possibility to recover solutions of Schr\"odinger equations from its time-averages in the setting where the values at the initial time are unknown. This problem can be reformulated as a new boundary value problem where a Cauchy condition is replaced by a prescribed time-average of the solution. It is shown that this new problem is well-posed in certain classes of solutions. The paper establishes existence, uniqueness, and a regularity of the solution for this new problem.