Effective Bounds for the Decay of Schrödinger Eigenfunctions and Agmon bubbles

2021/10/04 by Steinerberger, Stefan
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2110.01163

Abstract

We study solutions of -Δu + V u = λu on ℝⁿ. Such solutions localize in the `allowed' region \x ∈ ℝ<sup>n</sup>: V(x) ≤ λ\ and decay exponentially in the `forbidden' region \x ∈ ℝⁿ: V(x) &gt; 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.

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