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Localization of quantum states and landscape functions


Author: Stefan Steinerberger
Journal: Proc. Amer. Math. Soc. 145 (2017), 2895-2907
MSC (2010): Primary 35P20; Secondary 82B44
DOI: https://doi.org/10.1090/proc/13343
Published electronically: February 24, 2017
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Abstract: Eigenfunctions in inhomogeneous media can have strong localization properties. Filoche and Mayboroda showed that the function $ u$ solving $ (-\Delta + V)u = 1$ controls the behavior of eigenfunctions $ (-\Delta + V)\phi = \lambda \phi $ via the inequality

$\displaystyle \vert\phi (x)\vert \leq \lambda u(x) \Vert\phi \Vert _{L^{\infty }}.$

This inequality has proven to be remarkably effective in predicting localization and recently Arnold, David, Jerison, Mayboroda and Filoche connected $ 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 $ \phi (x)$ as an average over Brownian motion $ \omega (\cdot )$ started in $ x$

$\displaystyle \phi (x) = \mathbb{E}_{x}\left (\phi (\omega (t)) e^{\lambda t-\int _{0}^{t}{V(\omega (z))dz}} \right ).$

This variation estimate will guarantee that $ \phi $ has to change at least by a factor of 2 in a small ball, which implicitly creates a landscape whose relationship with $ 1/u$ we discuss.

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Additional Information

Stefan Steinerberger
Affiliation: Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, Connecticut 06511
Email: stefan.steinerberger@yale.edu

DOI: https://doi.org/10.1090/proc/13343
Keywords: Laplacian eigenfunction, localization, torsion function, Feynman-Kac
Received by editor(s): May 23, 2016
Published electronically: February 24, 2017
Communicated by: Svitlana Mayboroda
Article copyright: © Copyright 2017 American Mathematical Society