We recall some properties of convex functions and, in particular, of the sum of the largest eigenvalues of a Hermitian matrix. From these properties a new estimate of an arbitrary eigenvalue of a sum of Hermitian matrices is derived, which in turn is used to compute an approximate associated spectral projector. These estimates are applied for the first time to explain the generic spectral features of quantum systems. As an application of the formalism, we explain the preponderance of certain ground-state angular momenta as observed in the vibron model with random interactions. We show that the evolution of eigenstates can be predicted from the knowledge of a limited number of spectra and investigate the effect of a three-body interaction in the vibron model on eigenenergies and eigenvectors.