Effective bounds for the decay of Schrödinger eigenfunctions and Agmon bubbles

We study solutions of −Δu + Vu = λu on ℝⁿ. Such solutions localize in the ‘allowed’ region {x ∈ ℝⁿ: V(x) ≤ λ} and decay exponentially in the ‘forbidden’ region {x ∈ ℝⁿ: V(x) > λ}. One way of making this precise is Agmon’s inequality implying decay estimates in terms of the Agmon metric. We prove a complementary decay estimate in terms of harmonic measure which can improve on Agmon’s estimate, connect the Agmon metric to decay of harmonic measure and prove a sharp pointwise Agmon estimate.