  In this paper we develop and compare different real-time methods to calculate
spectral functions. These are classical-statistical simulations, the Gaussian
state approximation (GSA), and the functional renormalization group (FRG)
formulated on the Keldysh closed-time path. Our test-bed system is the quartic
anharmonic oscillator, a single self-interacting bosonic degree of freedom,
coupled to an external heat bath providing dissipation analogous to the
Caldeira-Leggett model. As our benchmark we use the spectral function from
exact diagonalization with constant Ohmic damping. To extend the GSA for the
open system, we solve the corresponding Heisenberg-Langevin equations in the
Gaussian approximation. For the real-time FRG, we introduce a novel general
prescription to construct causal regulators based on introducing
scale-dependent fictitious heat baths. Our results explicitly demonstrate how
the discrete transition lines of the quantum system gradually build up the
broad continuous structures in the classical spectral function as temperature
increases. At sufficiently high temperatures, classical, GSA and
exact-diagonalization results all coincide. The real-time FRG is able to
reproduce the effective thermal mass, but overestimates broadening and only
qualitatively describes higher excitations, at the present order of our
combined vertex and loop expansion. As temperature is lowered, the GSA follows
the ensemble average of the exact solution better than the classical spectral
function. In the low-temperature strong-coupling regime, the qualitative
features of the exact result are best captured by our real-time FRG
calculation, with quantitative improvements to be expected at higher truncation
orders.
