  We resolve both a conjecture and a problem of Z. Shen from the 90's regarding
non-asymptotic bounds on the eigenvalue counting function of the magnetic
Schr\"odinger operator $L_{{\bf a},V}=-(\nabla-i{\bf a})^2+V$ with a singular
or irregular magnetic field ${\bf B}$ on $\mathbb R^n$, $n\geq3$. We do this by
constructing a new landscape function for $L_{{\bf a},V}$, and proving its
corresponding uncertainty principle, under certain directionality assumptions
on ${\bf B}$, but with no assumption on $\nabla{\bf B}$. These results arise as
applications of our study of the Filoche-Mayboroda landscape function $u$, a
solution to the equation $L_Vu=-\operatorname{div} A\nabla u+Vu=1$, on
unbounded Lipschitz domains in $\mathbb R^n$, $n\geq1$, and $0\leq V\in
L^1_{\operatorname{loc}}$, under a mild decay condition on the Green's
function. For $L_V$, we prove a priori exponential decay of Green's function,
eigenfunctions, and Lax-Milgram solutions in an Agmon distance with weight
$1/u$, which may degenerate. Similar a priori results hold for $L_{{\bf a},V}$.
Furthermore, when $n\geq3$ and $V$ satisfies a scale-invariant Kato condition
and a weak doubling property, we show that $1/\sqrt u$ is pointwise equivalent
to the Fefferman-Phong-Shen maximal function $m(\cdot,V)$; in particular this
gives a strong scale-invariant Harnack inequality for $u$, and a setting where
the Agmon distance with weight $1/u$ is not too degenerate. Finally, we extend
results from the literature for $L_{{\bf a},V}$ regarding exponential decay of
the fundamental solution and eigenfunctions, to the situation of irregular
magnetic fields with directionality assumptions.
