  The Vlasov-Schr\"odinger-Poisson system is a kinetic-quantum hybrid model
describing quasi-lower dimensional electron gases. For this system, we
construct a large class of 2D kinetic/1D quantum steady states in a bounded
domain as generalized free energy minimizers, and we show their finite subband
structure, monotonicity, uniqueness and conditional dynamical stability. Our
proof is based on the concentration-compactness principle, but some additional
difficulties arise due to lack of compactness originated from the hybrid nature
(see Remark 1.9). To overcome the difficulties, we introduce a 3-step
refinement of a minimizing sequence by rearrangement and partial minimization
problems, and the coercivity lemma for the free energy (Lemma 5.3) is crucially
employed.
