  We present a new method of measuring the power spectrum of initial
perturbations to an unprecedently small scale of $\sim$ 10$h^{-1}$ kpc. We
apply this method to a sample of 4500 Ly-$\alpha$ absorbers and recover the
cold dark matter (CDM) like power spectrum at scales $\geq 300h^{-1}$kpc with a
precision of $\sim$ 10%. However at scales $\sim 10 - 300 h^{-1}$kpc the
measured and CDM--like spectra are noticeable different. This result suggests a
complex inflation with generation of excess power at small scales. The
magnitude and reliability of these deviations depend also upon the possible
incompleteness of our sample and poorly understood process of formation of weak
absorbers. Confirmation of the CDM--like shape of the initial power spectrum or
detection of its distortions at small scales are equally important for widely
discussed problems of physics of the early Universe, galaxy formation, and
reheating of the Universe. Our method links the observed mass function of
absorbers with the correlation function of the initial velocity field and
therefore it avoids the Nyquist restrictions limiting the investigations based
on the smoothed flux or density fields.
  The physical model of absorbers adopted here asserts that they are formed in
the course of both linear and nonlinear adiabatic or shock compression of dark
matter (DM) and gaseous matter. At scales $\geq 1h^{-1}$Mpc all characteristics
of the DM component and, in particular, their redshift distribution are found
to be consistent with theoretical expectations for Gaussian initial
perturbations with a CDM--like power spectrum.
