Localization of quantum states and landscape functions

Abstract

Eigenfunctions in inhomogeneous media can have strong localization properties. Filoche and Mayboroda showed that the function u u solving ( − Δ + V ) u = 1 (-\Delta + V)u = 1 controls the behavior of eigenfunctions ( − Δ + V ) ϕ = λ ϕ (-\Delta + V)\phi = \lambda \phi via the inequality \[ | ϕ ( x ) | ≤ λ u ( x ) ‖ ϕ ‖ L ∞ . |\phi (x)| \leq \lambda u(x) \|\phi \|_{L^{\infty }}. \] This inequality has proven to be remarkably effective in predicting localization and recently Arnold, David, Jerison, Mayboroda and Filoche connected 1 / u 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 ) \phi (x) as an average over Brownian motion ω ( ⋅ ) \omega (\cdot ) started in x x \[ ϕ ( x ) = E x ( ϕ ( ω ( t ) ) e λ t − ∫ 0 t V ( ω ( z ) ) d z ) . \phi (x) = \mathbb {E}_{x}\left (\phi (\omega (t)) e^{\lambda t-\int _{0}^{t}{V(\omega (z))dz}} \right ). \] This variation estimate will guarantee that ϕ \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 1/u we discuss.