X-ray photoionization of hydrogen in the presence of a bichromatic las
er held is considered. The expressions for the T matrix and the cross
sections for the laser-assisted x-ray photoionization are presented. T
he initial state is the laser-field-dressed hydrogen-atom ground state
, while the final state is the improved Coulomb-Volkov wave. The gauge
consistency is ensured by working in the r.E gauge. It turns out that
the matrix elements (the explicit form of which is given in the Appen
dix) are much simpler in this case. This enables one to consider the s
ymmetry properties of the T matrix and the cross sections analytically
. It was shown that the symmetry (phi+pi, pi-theta)<->(phi,theta), whe
re phi is the relative phase between the laser-field components and a
is the polar angle of the outgoing electron, is exact in our case. In
addition to this, there are approximate symmetries phi<->-phi and phi<
->phi+pi. All these symmetries, as well as the behavior of the differe
ntial and total cross sections as functions of phi, theta, and the num
ber of exchanged photons, are analyzed in numerous examples. The resul
ts presented show the possibility of the coherent phase control of the
laser-assisted x-ray photoionization process. It was also shown that
in the monochromatic case our model gives results that are in good agr
eement with the results of previous work.