  We study the spectral properties of Markovian driven-dissipative quantum
systems, focusing on the nonlinear quantum van der Pol oscillator as a
paradigmatic example. We discuss a generalized Lehmann representation, in which
single-particle Green's functions are expressed in terms of the eigenstates and
eigenvalues of the Liouvillian. Applying it to the quantum van der Pol
oscillator, we find a wealth of phenomena that are not apparent in the
steady-state density matrix alone. Unlike the steady state, the photonic
spectral function has a strong dependence on interaction strength. Further, we
find that the interplay of interaction and non-equilibrium effects can result
in a surprising "negative density of states", associated with a negative
temperature, and we identify two different mechanisms responsible for this
phenomenon.
