We study the model of two interacting particles moving in a 1D box, paying
main attention to the quantum-classical correspondence for the average shap
e of quantum eigenstates and for the local density of states (LDOS). We sho
w that if the classical motion is chaotic, in a deep semi-classical region
of a quantum system, both the shape of eigenstates and of the LDOS coincide
with their classical analogs, on average. However, individual eigenstates
exhibit quite large fluctuations which may not be treated as statistical on
es. Thus, comparison of quantum quantities to the classical ones allows one
to detect quantum effects of localization which for conservative systems e
merge in the energy space.