  We investigate almost-degenerate perturbation theory of eigenvalue problems,
using spectral projectors, also named density matrices. When several
eigenvalues are close to each other, the coefficients of the perturbative
series become singular because inverses of differences between eigenvalues
arise as some factors. We remove those artificial singularities in the
expressions of the coefficients of the series, allowing eigenvalue gaps to be
arbitrarily small and even vanishing in the resulting formulas.
