When computing the ideal gas thermal canonical partition function for a
    scalar outside a black hole horizon, one encounters the divergent
    single-particle density of states (DOS) due to the continuous nature of the
    normal mode spectrum. Recasting the Lorentzian field equation into an effective
    1D scattering problem, we argue that the scattering phases encode non-trivial
    information about the DOS and can be extracted by “renormalizing” the DOS with
    respect to a reference. This defines a renormalized free energy up to an
    arbitrary additive constant. Interestingly, the 1-loop Euclidean path integral,
    as computed by the Denef-Hartnoll-Sachdev formula, fixes the reference free
    energy to be that on a Rindler space, and the renormalized DOS captures the
    quasinormal modes for the scalar. We support these claims with the examples of
    scalars on static BTZ, Nariai black holes and the de Sitter static patch. For
    black holes in asymptotically flat space, the renormalized DOS is captured by
    the phase of the transmission coefficient whose magnitude squared is the
    greybody factor. We comment on possible connections with recent works from an
    algebraic point of view.

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