Localization of quantum states and landscape functions

Steinerberger, Stefan

doi:10.1090/proc/13343

Localization of quantum states and landscape functions
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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.

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