  By recasting the non-linear frequency-dependent $GW$ quasiparticle equation
into a linear eigenvalue problem, we explain the appearance of multiple
solutions and unphysical discontinuities in various physical quantities
computed within the $GW$ approximation. Considering the $GW$ self-energy as an
effective Hamiltonian, it is shown that these issues are key signatures of
strong correlation in the $(N\pm1)$-electron states and can be directly related
to the intruder state problem. A simple and efficient regularization procedure
inspired by the similarity renormalization group is proposed to avoid such
issues and speed up convergence of partially self-consistent $GW$ calculations.
