The density distributions of 10 Be and 11 Be nuclei obtained within the quantum Monte Carlo (QMC) model and the generator coordinate method (GCM) are used to calculate the microscopic optical potentials (OPs) and cross sections of elastic scattering of these nuclei on protons and 12 C at energies E < 100 MeV/nucleon. The real part of the OP is calculated using the folding model with the exchange terms included, while the imaginary part of the OP that reproduces the phase of scattering is obtained in the high-energy approximation (HEA). In this hybrid model of OP the free parameters are the depths of the real and imaginary parts obtained by fitting the experimental data. The well known energy dependence of the volume integrals is used as a physical constraint to resolve the ambiguities of the parameter values. The role of the spin-orbit potential and the surface contribution to the OP is studied for an adequate description of available experimental elastic scattering cross section data. Also, the cluster model, in which 11 Be consists of a n-halo and the 10 Be core, is adopted. Within the latter, the breakup cross sections of 11 Be nucleus on 9 Be, 93 Nb, 181 Ta, and 238 U targets and momentum distributions of 10 Be fragments are calculated and compared with the existing experimental data.