  The diagonal elements of the time correlation matrix are used to probe closed
quantum systems that are measured at random times. This enables us to extract
two distinct parts of the quantum evolution, a recurrent part and an
exponentially decaying part. This separation is strongly affected when spectral
degeneracies occur, for instance, in the presence of spontaneous symmetry
breaking. Moreover, the slowest decay rate is determined by the smallest energy
level spacing, and this decay rate diverges at the spectral degeneracies.
Probing the quantum evolution with the diagonal elements of the time
correlation matrix is discussed as a general concept and tested in the case of
a bosonic Josephson junction. It reveals for the latter characteristic
properties at the transition to Hilbert-space localization.
