  We investigate the quantitative unique continuation properties of real-valued
solutions to Schr\"odinger equations in the plane with potentials that exhibit
growth at infinity. More precisely, for equations of the form $\Delta u - V u =
0$ in $\mathbb{R}^2$, with $|V(z)| \lesssim |z|^{N}$ for some $N \ge 0$, we
prove that real-valued solutions satisfy exponential decay estimates with a
rate that depends explicitly on $N$. The case $N = 0$ corresponds to the Landis
conjecture, which was proved for real-valued solutions in the plane in
[LMNN20]. As such, the results in this article may be interpreted as
generalized Landis-type theorems. Our proof techniques rely heavily on the
ideas presented in [LMNN20].
