The cumulation of de Broglie waves of electrons in the hollow molecules (for example, C60) was investigated. The existence of negatively charged endo-ions of fullerenes with trapped electrons in the inner cavity (endo-electrons) is proved. The stationary equation Schrödinger (Helmholtz) for hollow polarized molecules is solved analytically. An endo-electron has a positive total energy, but because of the polarization forces is localized (constantly cumulating) in the center of the hollow spherically symmetric molecule (C60). Analytically calculated own ψ-function and own energy, determine the probability of finding an electron in a cavity of a hollow molecule. Proper energy of a quantum cavity (C60) is compared with experimental studies of cross sections of electron attachment to C60 dependencies on electron energy. The importance of the polarization hollow molecules in the stabilization of the fullerene endo-ions with endo-electrons (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 charge carriers, their thermal, electrical properties of semiconductors and hardening of materials with free electrons. Quantum properties of polarized cavities could cause a variation of parameters of nanocomposites with the concentration of quantum modifiers (C60).