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We examine the Liouvillian approach to the quantum Hall plateau transition,as introduced recently by Sinova, Meden, and Girvin [Phys. Rev. B {\bf 62},2008 (2000)] and developed by Moore, Sinova and Zee [Phys. Rev. Lett. {\bf 87},046801 (2001)]. We show that, despite appearances to the contrary, theLiouvillian approach is not specific to the quantum mechanics of particlesmoving in a single Landau level: we formulate it for a general disorderedsingle-particle Hamiltonian. We next examine the relationship betweenLiouvillian perturbation theory and conventional calculations ofdisorder-averaged products of Green functions and show that each term inLiouvillian perturbation theory corresponds to a specific contribution to thetwo-particle Green function. As a consequence, any Liouvillian approximationscheme may be re-expressed in the language of Green functions. We illustratethese ideas by applying Liouvillian methods, including their extension to $N_L> 1$ Liouvillian flavors, to random matrix ensembles, using numericalcalculations for small integer $N_L$ and an analytic analysis for large $N_L$.We find that behavior at $N_L > 1$ is different in qualitative ways from thatat $N_L=1$. In particular, the $N_L = \infty$ limit expressed using Greenfunctions generates a pathological approximation, in which two-particlecorrelation functions fail to factorize correctly at large separations of theirenergy, and exhibit spurious singularities inside the band of random matrixenergy levels. We also consider the large $N_L$ treatment of the quantum Hallplateau transition, showing that the same undesirable features are presentthere, too

Deconstructing the Liouvillian approach to the quantum Hall plateau transition