Conclusion

1. The first few eigenfunctions of the random Hamiltonian localize as P_{max} become big enough. This phenomenon is observed on all the domains we experimented on, i.e. triangle and square, Sierpinski gasket, Sierpinski carpet, random carpets, magic carpet and random magic carpets. For each of these domains, as P_{max} increases, the number of eigenfunctions that localize increases, starting from the lowest end of the spectrum; also the supports of the localized eigenfunctions become smaller.

2. Given Hu = 1 and eigenfunctions Hv = \lambda v, it is proven by Filoche and Mayboroda that |u|\ge \frac{|v|}{\lambda \|v\|_\infty} pointwise for any eigen pair v and \lambda, where \|v\|_\infty denote the supremum of |v| on the domain S. When P_{max} is big and localization happens, we observe that the cumulative level sets H_\epsilon = \{x\in\Omega|u(x)\le\epsilon\} outlines a cage within which the support of the first few eigenfunctions live. The peaks of the first few eigenfunctions, after aforesaid normalization, tend to occupy the space under peaks of u starting from the heighest to low. A trained eye can tell where the support of the first few eigenfunctions might live by looking at u. This phenomenon occurs for all the fractals.

3. The green function to the random Hamiltonian localizes as P_{max} increases. This happens for all domain.