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We study solutions of $-\Delta u + V u = \lambda u$ on $\mathbb{R}^n$. Suchsolutions localize in the `allowed' region $\left\{x \in \mathbb{R}^n: V(x)\leq \lambda\right\}$ and decay exponentially in the `forbidden' region$\left\{x \in \mathbb{R}^n: V(x) > \lambda\right\}$. One way of making thisprecise is Agmon's inequality implying decay estimates in terms of the Agmonmetric. We prove a complementary decay estimate in terms of harmonic measurewhich can improve on Agmon's estimate, connect the Agmon metric to decay ofharmonic measure and prove a sharp pointwise Agmon estimate.

Effective Bounds for the Decay of Schr\"odinger Eigenfunctions and Agmon bubbles