Abstract
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.
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Partially supported by the NSF (DMS-2123224) and the Alfred P. Sloan Foundation.
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Steinerberger, S. Effective bounds for the decay of Schrödinger eigenfunctions and Agmon bubbles. Isr. J. Math. (2024). https://doi.org/10.1007/s11856-024-2641-x
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DOI: https://doi.org/10.1007/s11856-024-2641-x