  We study a class of potentials $f$ on one sided full shift spaces over finite
or countable alphabets, called potentials of product type. We obtain explicit
formulae for the leading eigenvalue, the eigenfunction (which may be
discontinuous) and the eigenmeasure of the Ruelle operator. The uniqueness
property of these quantities is also discussed and it is shown that there
always exists a Bernoulli equilibrium state even if $f$ does not satisfy
Bowen's condition. We apply these results to potentials $f:\{-1,1\}^\mathbb{N}
\to \mathbb{R}$ of the form $$
  f(x_1,x_2,\ldots)
  =
  x_1 + 2^{-\gamma} \, x_2 + 3^{-\gamma} \, x_3 + ...+n^{-\gamma} \, x_n +
\ldots
  $$ with $\gamma >1$. For $3/2 < \gamma \leq 2$, we obtain the existence of
two different eigenfunctions. Both functions are (locally) unbounded and exist
a.s. (but not everywhere) with respect to the eigenmeasure and the measure of
maximal entropy, respectively.
