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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Localization of quantum states and landscape functions
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by Stefan Steinerberger PDF
Proc. Amer. Math. Soc. 145 (2017), 2895-2907 Request permission

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 \[ |\phi (x)| \leq \lambda u(x) \|\phi \|_{L^{\infty }}.\] This inequality has proven to be remarkably effective in predicting localization and recently Arnold, David, Jerison, Mayboroda and Filoche connected $1/u$ 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 $x$ \[ \phi (x) = \mathbb {E}_{x}\left (\phi (\omega (t)) e^{\lambda t-\int _{0}^{t}{V(\omega (z))dz}} \right ).\] 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.
References
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Additional Information
  • Stefan Steinerberger
  • Affiliation: Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, Connecticut 06511
  • MR Author ID: 869041
  • ORCID: 0000-0002-7745-4217
  • Email: stefan.steinerberger@yale.edu
  • Received by editor(s): May 23, 2016
  • Published electronically: February 24, 2017
  • Communicated by: Svitlana Mayboroda
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 2895-2907
  • MSC (2010): Primary 35P20; Secondary 82B44
  • DOI: https://doi.org/10.1090/proc/13343
  • MathSciNet review: 3637939