  The derivation of approximate wave functions for an electron submitted to
both a coulomb and a time-dependent laser electric fields, the so-called
Coulomb-Volkov (CV) state, is addressed. Despite its derivation for continuum
states does not exhibit any particular problem within the framework of the
standard theory of quantum mechanics (QM), difficulties arise when considering
an initially bound atomic state. Indeed the natural way of translating the
unperturbed momentum by the laser vector potential is no longer possible since
a bound state does not exhibit a plane wave form including explicitely a
momentum. The use of a fractal space permits to naturally define a momentum for
a bound wave function. Within this framework, it is shown how the derivation of
laser-dressed bound states can be performed. Based on a generalized eikonal
approach, a new expression for the laser-dressed states is also derived, fully
symmetric relative to the continuum or bound nature of the initial unperturbed
wave function. It includes an additional crossed term in the Volkov phase which
was not obtained within the standard theory of quantum mechanics. The
derivations within this fractal framework have highlighted other possible ways
to derive approximate laser-dressed states in QM. After comparing the various
obtained wave functions, an application to the prediction of the ionization
probability of hydrogen targets by attosecond XUV pulses within the sudden
approximation is provided. This approach allows to make predictions in various
regimes depending on the laser intensity, going from the non-resonant
multiphoton absorption to tunneling and barrier-suppression ionization.
