  We provide a simple theoretical model for the quasar luminosity function at
high redshifts that naturally reproduces the statistical properties of the
luminous SDSS quasar sample at redshifts z~4.3 and z>5.7. Our model is based on
the assumptions that quasar emission is triggered by galaxy mergers, and that
the black hole mass is proportional to a power-law in the circular velocity v
of the host galactic halo. We assume that quasars shine at their Eddington
luminosity over a time proportional to the mass ratio between the small and
final galaxies in the merger. This simple model fits the quasar luminosity
function at z~2-3, reproduces the normalization and logarithmic slope (beta
-2.58) at z~4.3, explains the space density of bright SDSS quasars at z~6.0,
reproduces the black hole - halo mass relation for dormant black holes in the
local universe, and matches the estimated duty cycle of quasar activity (~10^7
years) in Lyman-break galaxies at z~3. Based on the derived luminosity function
we predict the resulting gravitational lensing rates for high redshift quasars.
The lens fractions in the SDSS samples are predicted to be ~2% at z~4.3 and
~10% at z>5.7. Interestingly, the limiting quasar luminosity in our best-fit
relation of L proportional to v^5/G, scales as the binding energy of the host
galaxy divided by its dynamical time, implying that feedback is the mechanism
that regulates black hole growth in galactic potential wells.
