Localization of Quantum States and Landscape Functions

Eigenfunctions in inhomogeneous media can have strong localization properties. Filoche & Mayboroda showed that the function $u$ solving $(-\Delta

V)u = 1 $c o n t r o l s t h e b e h a v i o r o f e i g e n f u n c t i o n s$ (-\Delta + V)\phi = \lambda\phi $v i a t h e i n e q u a l i t y$ $|\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 \& 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 $\omega(\cdot)$ in started in $x$$ \phi(x) = $E$ {x}\left(\phi(\omega(t)) e^{\lambda t-\int_{0}^{t}{V(\omega(z))dz}} \right).$ $T h i s v a r i a t i o n e s t i m a t e w i l l g u a r a n t e e t h a t$ \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.

Eigenfunctions in inhomogeneous media can have strong localization properties. Filoche & Mayboroda showed that the function $u$ solving $(-\Delta

V)u = 1 $c o n t r o l s t h e b e h a v i o r o f e i g e n f u n c t i o n s$ (-\Delta + V)\phi = \lambda\phi $v i a t h e i n e q u a l i t y$ $|\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 \& 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 $\omega(\cdot)$ in started in $x$$ \phi(x) = $E$ {x}\left(\phi(\omega(t)) e^{\lambda t-\int_{0}^{t}{V(\omega(z))dz}} \right).$ $T h i s v a r i a t i o n e s t i m a t e w i l l g u a r a n t e e t h a t$ \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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