Ivan Veselić
Technical University of Chemnitz, Faculty of Mathematics
Spectral averaging in the mathematical theory of Anderson localization
Vrijeme: srijeda, 14. 3. 2012., 14:15 sati (točno)
Mjesto: Fizički odsjek, Bijenička c. 32, predavaonica F201
Anderson localization is a widely studied topic in theoretical and computational physics. It is also an active field of research in mathematics, even though only a part of the results suggested by physical arguing can be proven (or in rare occasions disproven) on the mathematical level of rigour.
Spectral averaging is the pheonomenon that certain types of disorder regularize and smooth out spectral data, like the density of states. It implies that resonances on large scales can occur only with small probability, thus being crucial for the understanding of the localization of eigenfunctions. It also allows to prove that eigenvalues are non-stationary w.r.t. fluctuations of the disorder
We discuss several features of the underlying random Hamiltonian which play a key role in the rigourous mathematical analysis of Anderson localization and have an appealing physical meaning, e.g.
- continuously distributed random variables, versus Bernoulli distributetd ones: For the latter, spectral averaging is harder to prove, in fact the density of states need not be smooth
- positive definite versus semindefinite random perturbations:
For the latter it is not clear whether certain spectral subspaces remain unaffected by the disorder.
- monotone versus non-monotone random perturbations:
For the latter it is not clear how to quantitatively estimate the movement or flow of eigenvalues.
Hrvoje Buljan, hbuljan@phy.hr
Pristupačnost
