The cumulation of de Broglie waves of electrons in hollow molecules (for example, C60) has been investigated. The existence of negatively charged endoions of fullerenes with trapped electrons in the inner cavity (endoelectrons) is proved. The Schrodinger (Helmholtz) stationary equation for hollow polarized molecules is solved analytically. An endoelectron has a positive total energy, but, because of the polarization forces, it is localized (constantly cumulating) in the center of the hollow spherically symmetric molecule (C60). The analytically calculated Eigen ψ function and eigen energy determine the probability of finding an electron in the cavity of a hollow molecule. The Eigen energy of a quantum cavity (C60) is compared with ons attachment to C60 dependencies on the electron energy. The importance of the polarization hollow molecules in the stabilization of the fullerene endoions with endoelectrons (with energies from 0.2 to 12 eV) is proved. The effect of the cumulation of electrons in the hollow of the molecule (the trap for electrons) can be used to control the concentration of the charge carriers, the thermal and electrical properties of semiconductors, and the hardening of materials with free electrons. The quantum properties of polarized cavities could cause the variation of the parameters of nanocomposites with a concentration of quantum modifiers (C60).