  It is generally accepted that the dynamical mean field theory gives a good
solution of the Holstein model, but only in dimensions greater than two. Here,
we show that this theory, which becomes exact in the weak coupling and in the
atomic limit, provides an excellent, numerically cheap, approximate solution
for the spectral function of the Holstein model in the whole range of
parameters, even in one dimension. To establish this, we make a detailed
comparison with the spectral functions that we obtain using the newly developed
momentum-space numerically exact hierarchical equations of motion method, which
yields electronic correlation functions directly in real time. We crosscheck
these conclusions with our path integral quantum Monte Carlo and exact
diagonalization results, as well as with the available numerically exact
results from the literature.
