  We discuss a quantum mechanical indirect measurement method to recover a
position dependent Hamilton matrix from time evolution of coherent quantum
mechanical states through an object. A mathematical formulation of this inverse
problem leads to weighted X-ray transforms where the weight is a matrix. We
show that such X-ray transforms are injective with very rough weights.
Consequently, we can solve our quantum mechanical inverse problem in several
settings, but many physically relevant problems we pose also remain open. We
discuss the physical background of the proposed imaging method in detail. We
give a rigorous mathematical treatment of a neutrino tomography method that has
been previously described in the physical literature.
