Keldysh theory re-examined

A derivation of the ionization rate for a hydrogen atom in its ground state (or a hydrogen-like positive ion) in a strong linearly polarized laser field is presented. The derivation utilizes the famous Keldysh probability amplitude in the length gauge (in the dipole approximation) and without Coulomb effects in the final state of the ionized electron. No further approximations are made, because the amplitude has been expanded in the double Fourier series in a time domain (with the help of the generalized Bessel functions). Thus, our theory has no other limitations that are characteristic of the original Keldysh theory. We compare our 'exact' theory with the original Keldysh one by studying photoionization energy spectra and total ionization rates. We show a breakdown of the original Keldysh theory for higher frequencies (when the photon energy approaches the binding energy). We also compare our theory with the analogous result in the velocity gauge. In the barrier-suppression regime, the 'exact' Keldysh theory gives results which are close to the well-known empirical formula and close to some other numerical or theoretical results. Numerous comparisons of total ionization rates are limited to photons of energies lower or much lower than the binding energy of the atom. (paper)