Effects of gravity in extra dimensions in atomic phenomena

We use the difference between theory and experiment for energy intervals in simple atomic systems (hydrogen, muonium, positronium and deuteron) to find limits on the size of extra space dimensions in the Arkani-Hamed - Dimopoulos - Dvali model for gravitation potential on short distances. As an additional experimental fact we use absence of the small size gravitational bound states of elementary particles. We demonstrate that the perturbation theory approach does not work and more reliable results are obtained by solving the Dirac equations for an electron in Coulomb and gravitational fields. These results probe smaller distances than distance between nuclei in molecules and the limits are significantly stronger than the limits on the size of extra dimensions obtained using spectra of hydrogen molecules.