  We study solutions of the collisionless Boltzmann equation (CBE) in a
functional Koopman representation. This facilitates the use of linear spectral
techniques characteristic of the analysis of Schrodinger-type equations. For
illustrative purposes, we consider the classical phase mixing of a
non-interacting distribution function in a quartic potential. Solutions are
determined perturbatively relative to a harmonic oscillator. We impose a form
of coarse-graining by choosing a finite dimensional basis to represent the
distribution function and time evolution operators, which sets a minimum length
scale on phase space structure. We observe a relationship between the dimension
of the representation and the multiplicity of the harmonic oscillator
eigenvalues. System dynamics are understood in terms of degenerate subspaces of
the linear operator spectra. Each subspace is associated with a mode of the
harmonic oscillator, the first two being bending and breathing structures. The
quartic potential splits the degenerate eigenvalues within each subspace. This
facilitates the formation of spiral structure as deformations from the harmonic
oscillator modes. We ultimately argue that this construction provides a
promising avenue for study of self-interacting systems experiencing phase
mixing, which is an outstanding problem in the context of the Gaia DR2 vertical
phase space spirals.
