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

TL;DR

This paper establishes new bounds on the exponential decay of Schrödinger eigenfunctions, improving existing estimates by connecting Agmon's metric with harmonic measure decay, and provides sharp pointwise decay bounds.

Contribution

It introduces a harmonic measure-based decay estimate that enhances Agmon's inequality, linking geometric and probabilistic methods for Schrödinger eigenfunction decay.

Findings

Harmonic measure decay bounds can improve Agmon's estimates.

Established a connection between Agmon metric and harmonic measure decay.

Proved a sharp pointwise decay estimate for eigenfunctions.

Abstract

We study solutions of $-\Delta u + V u = \lambda u$ on $\mathbb{R}^n$. Such solutions 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 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.