  The method proposed by the present authors to deal analytically with the
problem of Anderson localization via disorder [J.Phys.: Condens. Matter {\bf
14} (2002) 13777] is generalized for higher spatial dimensions D. In this way
the generalized Lyapunov exponents for diagonal correlators of the wave
function, $<\psi^2_{n,\mathbf{m}}>$, can be calculated analytically and
exactly. This permits to determine the phase diagram of the system. For all
dimensions $D > 2$ one finds intervals in the energy and the disorder where
extended and localized states coexist: the metal-insulator transition should
thus be interpreted as a first-order transition. The qualitative differences
permit to group the systems into two classes: low-dimensional systems ($2\leq D
\leq 3$), where localized states are always exponentially localized and
high-dimensional systems ($D\geq D_c=4$), where states with non-exponential
localization are also formed. The value of the upper critical dimension is
found to be $D_0=6$ for the Anderson localization problem; this value is also
characteristic of a related problem - percolation. Consequences for numerical
scaling and other approaches are discussed in detail.
