  The Schr\"odinger-Langevin (SL) equation is considered as an effective open
quantum system formalism suitable for phenomenological applications involving a
quantum subsystem interacting with a thermal bath. We focus on two open issues
relative to its solutions: the stationarity of the excited states of the
non-interacting subsystem when one considers the dissipation only and the
thermal relaxation toward asymptotic distributions with the additional
stochastic term. We first show that a proper application of the Madelung/polar
transformation of the wave function leads to a non zero damping of the excited
states of the quantum subsystem. We then study analytically and numerically the
SL equation ability to bring a quantum subsystem to the thermal equilibrium of
statistical mechanics. To do so, concepts about statistical mixed states and
quantum noises are discussed and a detailed analysis is carried with two kinds
of noise and potential. We show that within our assumptions the use of the SL
equation as an effective open quantum system formalism is possible and discuss
some of its limitations.
