  Using the Hubble parameter as new `inverse time' coordinate ($H$-formalism),
a new method of reconstructing the inflaton potential is developed also using
older results which, in principle, is applicable to any order of the slow-roll
approximation. In first and second order, we need three observational data as
inputs: the scalar spectral index $n_s$ and the amplitudes of the scalar and
the tensor spectrum. We find constraints between the values of $n_s$ and the
corresponding values for the wavelength $\lambda $. By imposing a dependence
$\lambda (n_s)$, we were able to reconstruct and visualize inflationary
potentials which are compatible with recent COBE and other astrophysical
observations. >From the reconstructed potentials, it becomes clear that one
cannot find only one special value of the scalar spectral index $n_s$.
