  An inhomogeneous (1+1)-dimensional model of the quantum gravity is
considered. It is found, that this model corresponds to a string propagating
against some curved background space. The quantization scheme including the
Wheeler-DeWitt equation and the "particle on a sphere" type of the gauge
condition is suggested. In the quantization scheme considered, the "problem of
time" is solved by building of the quasi-Heisenberg operators acting in a space
of solutions of the Wheeler-DeWitt equation and the normalization of the wave
function corresponds to the Klein-Gordon type. To analyze the physical
consequences of the scheme, a (1+1)-dimensional background space is considered
for which a classical solution is found and quantized. The obtained estimations
show the way to solution of the cosmological constant problem, which consists
in compensation of the zero-point oscillations of the matter fields by the
quantum oscillations of the scale factor. Along with such a compensation, a
slow global evolution of a background corresponding to an universe expansion
exists.
