  Signatures of the Coulomb corrections in the photoelectron momentum
distribution during laser-induced ionization of atoms or ions in tunneling and
multiphoton regimes are investigated analytically in the case of an
one-dimensional problem. High-order Coulomb corrected strong-field
approximation is applied, where the exact continuum state in the S-matrix is
approximated by the eikonal Coulomb-Volkov state including the second-order
corrections to the eikonal. Although, without high-order corrections our theory
coincides with the known analytical R-matrix (ARM) theory, we propose a
simplified procedure for the matrix element derivation. Rather than matching
the eikonal Coulomb-Volkov wave function with the bound state as in the
ARM-theory to remove the Coulomb singularity, we calculate the matrix element
via the saddle-point integration method as by time as well as by coordinate,
and in this way avoiding the Coulomb singularity. The momentum shift in the
photoelectron momentum distribution with respect to the ARM-theory due to
high-order corrections is analyzed for tunneling and multiphoton regimes. The
relation of the quantum corrections to the tunneling delay time is discussed
