Year: 2016
Type: article
Abstract: Eigenfunctions in inhomogeneous media can have strong localization properties. Filoche and Mayboroda showed that the function u u solving ( − Δ + V ) u = 1 (-\Delta + V)u = 1 controls the behavior of eigenfunctions ( − Δ + V ) ϕ = λ ϕ (-\Delta + V)\phi = \lambda \phi via the inequality \[ | ϕ ( x ) | ≤ λ u ( x ) ‖ ϕ ‖ L ∞ . |\phi (x)| \leq \lambda u(x) \|\phi \|_{L^{\infty }}. \] This inequality has proven to be remarkably effective in predicting localization and recently Arnold, David, Jerison, Mayboroda and Filoche connected 1 / u 1/u to decay properties of eigenfunctions. We aim to clarify properties of the landscape: the main ingredient is a localized variation estimate obtained from writing ϕ ( x ) \phi (x) as an average over Brownian motion ω ( ⋅ ) \omega (\cdot ) started in x x
DOI: 10.1090/proc/13343
Author: Stefan Steinerberger
Institution: Yale University
Language: Latvian
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2256247477