ANU Open Research Repository has been upgraded. We are still working on a few minor issues, which may result in short outages throughout the day. Please get in touch with if you experience any issues.

Localisation of Low Energy Eigenfunctions to the Schr ?odinger Operator




Zhang, Wenqi

Journal Title

Journal ISSN

Volume Title



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.






Thesis (Honours)

Book Title

Entity type

Access Statement

License Rights



Restricted until