For random Schrödinger potentials, the full law of the DN map determines the law of the potential, but its expectation or finite moments do not determine the mean; the averaged Green's operator determines mean and variance, and all moments in a two-atom model.
Determining a random Schr¨ odinger equation with unknown source and potential.SIAM Journal on Mathematical Analysis, 51(4):3465–3491, 2019
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A note on several inverse problems with generally random coefficients
For random Schrödinger potentials, the full law of the DN map determines the law of the potential, but its expectation or finite moments do not determine the mean; the averaged Green's operator determines mean and variance, and all moments in a two-atom model.