Effective Bounds for the Decay of Schrödinger Eigenfunctions and Agmon bubbles

Abstract

We study solutions of $-\Delta u+Vu=\lambda u$ on ${\mathbb{R}}^n$. Such solutions localize in the `allowed' region $\left\{x\in {\mathbb{R}}^n:V(x)\le \lambda \right\}$ and decay exponentially in the `forbidden' region $\left\{x\in {\mathbb{R}}^n:V(x)>\lambda \right\}$. 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.

References in corpus (1)

• Configurations of balls in Euclidean space that Brownian motion cannot avoid