  We review in details a recently proposed technique to extract information
about dynamical correlation functions of many-body hamiltonians with a few
Lanczos iterations and without the limitation of finite size. We apply this
technique to understand the low energy properties and the dynamical spectral
weight of a simple model describing the motion of a single hole in a quantum
antiferromagnet: the $t-J_z$ model in two spatial dimension and for a double
chain lattice. The simplicity of the model allows us a well controlled
numerical solution, especially for the two chain case. Contrary to previous
approximations we have found that the single hole ground state in the infinite
system is continuously connected with the Nagaoka fully polarized state for
$J_z \to 0$. Analogously we have obtained an accurate determination of the
dynamical spectral weight relevant for photoemission experiments. For $J_z=0$
an argument is given that the spectral weight vanishes at the Nagaoka energy
faster than any power law, as supported also by a clear numerical evidence. It
is also shown that spin charge decoupling is an exact property for a single
hole in the Bethe lattice but does not apply to the more realistic lattices
where the hole can describe closed loop paths.
