  Let $\displaystyle L = -\frac{1}{w} \, \mathrm{div}(A \, \nabla u) + \mu$ be
the generalized degenerate Schr\"odinger operator in $L^2_w(\mathbb{R}^d)$ with
$d\ge 3$ with suitable weight $w$ and measure $\mu$. The main aim of this paper
is threefold. First, we obtain an upper bound for the fundamental solution of
the operator $L$. Secondly, we prove some estimates for the heat kernel of $L$
including an upper bound, the H\"older continuity and a comparison estimate.
Finally, we apply the results to study the maximal function characterization
for the Hardy spaces associated to the critical function generated by the
operator $L$.
