Real-time methods for spectral functions

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