Random Schrödinger Operators
Over the last 30 years, the investigation of self-adjoint random operators turned into a fruitful area of mathematics,
which connects functional analysis and probability.
Since the effect of randomness on spectral and dynamical properties are profound and largely universal, this area offers some major challenges for analysts.
One of them is the extended states/localization conjecture. Perturbing a local operator by randomness,
the resulting spectrum is predicted to comprise a regimes with localized and
another one with extended states.
While the first regime is now understood fairly well in mathematical terms,
our understanding of the extended states regime is still underdeveloped.
In its weakest form, the extended states conjecture refers to a proof of the stability of some ac spectrum under weak, but
homogeneous randomness in more than two space dimensions.
Our own research has been focusing on the case of "infinite dimensions", namely tree graphs.
Another questions relates to a universality conjecture in random matrix theory: it is widely believed that
the spectral properties are related to the local statistics of the eigenvalues of a random Schr"odinger operator on a finite domain. Whereas localized states are proven to yield Poisson statistics, extended states are believed to be give rise to one of the classical ensembles in random matrix theory.
Among the current research topics is also the fate of the localization regime for a system of many interacting particles.
- M. Aizenman, S. Warzel: Localization bounds for multiparticle systems, http://arxiv.org/abs/0809.3436v1
- M. Aizenman, S. Warzel, The canopy graph and level statistics for random operators on trees, Math. Phys. Anal. Geom. 9: 291-333 (2007)
- M. Aizenman, B. Sims, S. Warzel, Stability of the absolutely continuous spectrum of random Schrödinger operators on tree graphs, Probab. Theory Relat. Fields 136: 363-394 (2006)
- H. Leschke, P. Müller, S. Warzel, A survey of rigorous results on random Schrödinger operators for amorphous solids, Markov Proc. Relat. Fields 9: 729-760 (2003)