  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.
