14 July to 25 July 2008

Isaac Newton Institute for Mathematical Sciences, Cambridge, UK

**Organiser:** Martin Zirnbauer (*Cologne*)

in association with the Newton Institute programme Mathematics and Physics of Anderson localization: 50 Years After (14 July to 19 December 2008)

Programme | Participants | Application | Accommodation and Cost | Accepted Posters | Photograph | Web Seminars

When a single-particle quantum Hamiltonian system is subjected to a
disorder potential, it is expected on physical grounds that a
transition from localised to extended energy eigenstates takes place
as a function of the disorder strength. Such a transition should be
accompanied by a characteristic change in the energy spectrum: if the
disorder is large enough for Anderson localisation to occur, the
random Schrödinger operator is known to have dense point spectrum;
on the other hand, if the disorder is weak and the space dimension
larger than *d* = 2, then one expects the existence of absolutely
continuous spectrum.

Giving a mathematical proof of this conjectured scenario, and
clarifying the nature of the spectrum and the eigenfunctions at the
transition point or in *d* = 2, remains an important and outstanding
problem of mathematical physics. Many features of the scenario are
believed to extend to a broader class of quantum systems including,
most prominently, those exhibiting transitions of Quantum Hall type.

This training course is mainly directed at researchers in early stages of their careers. Its aim is to provide the participants with an introduction to the subject, by exposing them to ideas, terminology and analytical techniques of the rigorous as well as the heuristic kind. Methods used in the study of Anderson localisation by mathematicians and by theoretical physicists will be reviewed by experts from both communities. Reviewing the state of the art for both disciplines will hopefully help to bridge the existing language gap between the communities and create an environment conducive to fruitful collaboration between physicists and mathematicians during the rest of the program.

- phenomenology of Anderson localisation (T. Spencer)
- introduction to the spectral theory of random Schrödinger operators (L. Pastur)
- introduction to supermatrix techniques and the nonlinear σ-model (Y. Fyodorov)
- rigorous techniques for 1
*D*and quasi 1*D*systems (I. Goldsheid) - rigorous methods in the statistical mechanics of phase transitions (D. Brydges)
- critical phenomena in two-dimensional disordered systems (A. Ludwig)

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