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Reconstruction of Hamiltonians from given time evolutions

In this paper, we propose a systematic method to solve the inverse dynamical problem for a finite-level quantum system governed by the von Neumann equation: to find a class of Hamiltonians reproducing a prescribed time evolution of a pure or mixed state of the system. Our approach exploits the equivalence between the action of the group of evolution operators over the state space and the adjoint action of the unitary group over Hermitian matrices. The method is illustrated by two examples involving a pure and a mixed state.