  The direct transition-matrix approach to the description of the electric
polarization of the quantum bound system of particles is used to determine the
electric multipole polarizabilities of the hydrogen-like atoms. It is shown
that in the case of the bound system formed by the Coulomb interaction the
corresponding inhomogeneous integral equation determining an off-shell
scattering function, which consistently describes virtual multiple scattering,
can be solved exactly analytically for all electric multipole polarizabilities.
Our method allows to reproduce the known Dalgarno-Lewis formula for electric
multipole polarizabilities of the hydrogen atom in the ground state and can
also be applied to determine the polarizability of the atom in excited bound
states.
