  Collisional reservoirs are becoming a major tool for modelling open quantum
systems. In their simplest implementation, an external agent switches on, for a
given time, the interaction between the system and a specimen from the
reservoir. Generically, in this operation the external agent performs work onto
the system, preventing thermalization when the reservoir is at equilibrium. One
can recover thermalization by considering an autonomous global setup where the
reservoir particles colliding with the system possess a kinetic degree of
freedom. The drawback is that the corresponding scattering problem is rather
involved. Here, we present a formal solution of the problem in one dimension
and for flat interaction potentials. The solution is based on the transfer
matrix formalism and allows one to explore the symmetries of the resulting
scattering map. One of these symmetries is micro-reversibility, which is a
condition for thermalization. We then introduce two approximations of the
scattering map that preserve these symmetries and, consequently, thermalize the
system. These relatively simple approximate solutions constitute models of
quantum thermostats and are useful tools to study quantum systems in contact
with thermal baths. We illustrate their accuracy in a specific example, showing
that both are good approximations of the exact scattering problem even in
situations far from equilibrium. Moreover, one of the models consists of the
removal of certain coherences plus a very specific randomization of the
interaction time. These two features allow one to identify as heat the energy
transfer due to switching on and off the interaction. Our results prompt the
fundamental question of how to distinguish between heat and work from the
statistical properties of the exchange of energy between a system and its
surroundings.
