Let
$x \in \mathbb {R}^{d}$
, *d* ≥ 3, and
$f: \mathbb {R}^{d} \rightarrow \mathbb {R}$
be a twice differentiable function with all second partial derivatives being continuous. For 1 ≤ *i*, *j* ≤ *d*, let
$a_{ij} : \mathbb {R}^{d} \rightarrow \mathbb {R}$
be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrödinger operator associated to
$$\mathcal{L}f(x) = \frac12 \sum\limits_{i=1}^{d} \sum\limits_{j=1}^{d} \frac{\partial}{\partial x_{i}} \left( a_{ij}(\cdot) \frac{\partial f}{\partial x_{j}}\right)(x) + {\int}_{\mathbb{R}^{d}\setminus{\{0\}}} [f(y) - f(x) ]J(x,y)dy $$
where
$J: \mathbb {R}^{d} \times \mathbb {R}^{d} \rightarrow \mathbb {R}$
is a symmetric measurable function. Let
$q: \mathbb {R}^{d} \rightarrow \mathbb {R}.$
We specify assumptions on *a*, *q*, and *J* so that non-negative bounded solutions to
$$\mathcal{L}f + qf = 0 $$
satisfy a Harnack inequality. As tools we also prove a Carleson estimate, a uniform Boundary Harnack Principle and a 3G inequality for solutions to
$\mathcal {L}f = 0.$