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Abstract:
This special issue captures recent developments in the classical phase problem in quantum mechanics, optics, and related areas. In particular, Born's rule in quantum mechanics from 1926, for which he was awarded the Nobel Prize in 1954, states that the square of the amplitude of a particle's wave function is proportional to the probability of finding the particle at a given point, whereas the wave function itself has no direct physical interpretation.
Similarly, most optical measurement techniques only provide information on the amplitude, but not on the phase of time-harmonic electromagnetic waves. The reason is that in many important cases of monochromatic electromagnetic wave propagation the wave frequency is so great that only the field intensity can be measured by modern technical devices.
While classical inverse scattering theory has focused mostly on data with phase information, new exciting results on phaseless data have appeared recently on the theoretical, algorithmic, and experimental side. As two particularly successful experimental techniques which lead to closely related mathematical models involving phase retrieval, we mention phase contrast x-ray imaging and cryo-electron microscopy. The latter was recognized by the Nobel Prize in Chemistry to Dubochet, Frank and Henderson in 2017; see [5, 16].
The twelve papers collected in this special issue focus on different aspects of phaseless inverse problems and provide a good overview on the state-of the-art of this active field of research. They may be grouped into four areas involving three papers each.
