Localisation of Low Energy Eigenfunctions to the Schr ?odinger Operator

Abstract

We explore the phenomena where low energy eigenfunctions of the operator L = - d + V for V 2 L1 concentrate on a small subset of the original domain, and decay exponentially outside of this region. We consider the function u solving the equation Lu = 1 and show that 1 u acts as an e ective potential, in the sense that one can show Z jrfj2 + V f2 = Z u2jr(f=u)j2 + 1 u f2: With this equality, it can be shown that the regions where 1 u < for some (small) eigenvalue approximately corresponds to where the corresponding eigenfunction is localised. Furthermore, if the regions where 1 u < are composed of suitably disjoint subsets, we show that if a (global) eigenfunction with eigenvalue resides in one of these subsets, then it is well approximated in this subset by local eigenfunctions (i.e. eigenfunctions of L in this subset) with eigenvalue . For these subsets to exist, 1 u needs to vary substantially. Since u solves Lu = 1, this phenomena cannot occur when V is smooth, and does not vary substantially. As a visual aid to gain intuition on how V a ects the localisation, we simulate the lowest energy eigenfunction to L numerically. We generate V to take random values on a dyadic grid. We vary the maximum size of the random values and the scale of the grid. Once the grid becomes su ciently ne, and for large values of V we observe that the lowest energy eigenfunction become concentrated in one particular region. We conjecture that the maximum size of V has a greater e ect on localisation than the grid size. This is also hinted at since the theorems depend only on jjV jj1. However, further work needs to be done for fractal potentials V , in the sense that the set of discontinuities of our V may have the same Hausdor dimension (no matter the grid size). Finally, we also explore brie y the possibilities of de ning a Brownian motion on endowed with a distance associated with 1 u. The hope is to one day be able to have a microscopic (Feyynman-Kac) description of the solution to @tu = Lu.We only explore a few possibilities to de ne an adapted Brownian motion on , we leave an explicit construction and the Feynman-Kac analysis for future work.