Graduate Seminar (S4B2-GS PDE)

Random Schrödinger operators and localization.

 

Information meeting (via zoom)  Thursday February 11 at 12h15

for the zoom link please send me an email  disertori(at)iam.uni-bonn.de

If you cannot come to the information meeting, but you are interested in the seminar, please contact me by email

 

 

Description.

It is a well known fact that in certain materials, such as alloys and amorphous media, the presence of disorder may destroy the conductivity properties and induce a transition between a conducting phase (at weak disorder) and an insulating  phase (at strong disorder). This phenomenon  is called the Anderson localization. 

Mathematically, the problem can be modeled by solutions of the following (stochastic) PDE

                      i∂t u= Hu    where  (Hu)(x)= -(Δu)(x)+ V(x) u(x)

where ∆ is the Laplacian operator (may be on Zd or Rd ) and {V(x)}x  is a family of random variables. Much useful and precise information can be gained using relatively elementary, but mathematically rigorous, methods. These rely on the study of the resolvant  GE (x,y):= (E Id -H)-1(x,y). More precisely one considers the average E[|GE(x, y)|s ], for certain values of s. In this seminar we plan to learn some of the most famous tools in this context.

Mostly we will rely on an introductory paper by Günter Stolz:
"An Introduction to the Mathematics of Anderson Localization" (see below)

Prerequisites. Functional analysis. Some basic knowledge in probability may be useful but not necessary.

 

Reading material:

  • "Self-adjoint operators and solving the Schroedinger equation" , Günter Stolz, lectures given at NSF/CBMS Conference on Quantum Spin Systems June 16-20, 2014 University of Alabama at Birmingham, pdf file available here                                              
  • ''An Introduction to the Mathematics of Anderson Localization” , Günter Stolz, in ”Contemporary Mathematics Vol. 552 2011”, preprint: arXiv:1104.2317                                                                                                                                                                 
  • "A short introduction to Anderson localization" , Dirk Hundertmark, in proceedings of the LMS Meeting on Analysis and Stochastics of Growth Processes and Interface Models, Bath, September 2006. Oxford Univ. Press, Oxford, 2008, 194-218, preprint available on his webpage here                                                                                                                                           
  • "An invitation to Random Schroedinger Operators",   Werner Kirsch, in: Panor. Synthèses 25, 1-119 (2008), preprint available here                                                                                                                                                                                                       
  • "How large is large? Estimating the critical disorder for the Anderson model" , Jeffrey Schenker Lett. Math. Phys., 105 (2015), 1-9, preprint: arXiv:1305.6987                                                                                                                                                         
  • "Finite volume fractional moment criteria for Anderson localization", M. Aizenman, J. Schenker, R. Friedrich and D. Hundertmark Comm. Math. Phys. 224 (2001), 219–253, preprint available here                                                                            
  • "Localization at large disorder and at extreme energies: an elementary derivation", M. Aizenman and S. Molchanov, Comm. Math. Phys. 157 (1993), 245–278

      

 


  

  

 

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