Localization of Quantum States and Landscape Functions

Eigenfunctions in inhomogeneous media can have strong localization properties. Filoche \& Mayboroda showed that the function $u$ solving $(-Δ+ V)u = 1$ controls the behavior of eigenfunctions $(-Δ+ V)ϕ= λϕ$ via the inequality $|ϕ(x)| \leq λu(x) \|ϕ\|_{L^{\infty}}.$ 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 $ϕ(x)$ as an average over Brownian motion $ω(\cdot)$ in started in $x$ $ϕ(x) = \mathbb{E}_{x}\left(ϕ(ω(t)) e^{λt-\int_{0}^{t}{V(ω(z))dz}} \right).$ This variation estimate will guarantee that $ϕ$ 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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