The quasi-normal modes of relativistic Fokker-Planck kinetic theory

Employing the well-known unitary equivalence between Fokker-Planck operators and Schrödinger Hamiltonians, we compute the quasi-normal-mode spectrum of ultrarelativistic kinetic theories with momentum-space diffusion.

We show that the collision operator reduces to a Dirac-delta Schrödinger problem in one spatial dimension, and to a Coulomb Schrödinger operator with hydrogenic spectrum in three dimensions.

Finite spatial wavenumber appears as a perturbation of the associated quantum potential.

The hydrodynamic mode is found to obey exact Fick-type diffusion at all real wavenumbers, whereas relativistic kinematics generically produces a continuous ballistic band in the non-hydrodynamic sector, a feature absent in the Newtonian regime.