1. Benasque, August 31, 2012 http://picasaweb.google.com/felix.izrailev Photo made during the conference can be found on the web page (see album “Benasque”):

2. Benasque, August 31, 2012 From closed to open 1D Anderson model:Transport versus spectral statistics F.M.Izrailev Instituto de Física, BUAP, Puebla, México and Michigan State University, USA V. G. Zelevinsky (NSCL, E.Lansing, USA) In collaboration with: S. Sorathia (IFUAP, Puebla, Mexico), G.Celardo (Univesita di Cattolica, Brescia, Italy),

3. Benasque, August 31, 2012 Open 1D Anderson model Ideal semi-infinite LEFT lead Ideal semi-infinite RIGHT lead How scattering properties depend on the degree of internal chaos and coupling strength ? Published in Phys. Rev. E (2012)

4. Benasque, August 31, 2012

6. Level statistics Spacing between neighbouring eigenvalues Spacing normalized to mean local spacing Level spacing distribution- provides information about the degree of chaos Ensemble average combined with average over eigenvalues that lie within a small energy window at the band centre Regular picket fence Chaotic Wigner-Dyson Localized Poisson Phenomenological spacing distribution interpolates between all three regimes 2D Coulomb gas on a circle repulsion parameter

7. Benasque, August 31, 2012 2D Coulomb gas on a circle F.M.Izrailev, 1990-91 F.Dyson, 1962

8. Benasque, August 31, 2012 Distribution of spacings between eigenvalues in the 1D Anderson model is the same as that for spacings between classical charged particles in the Coulomb gas model ! Chaotic Poisson Delta function

9. Benasque, August 31, 2012 Repulsion parameter is the properly normalized localization length !

10. Benasque, August 31, 2012 Non-Hermitian Effective Hamiltonian using the identity The exact non-Hermitian effective Hamiltonian Near the centre of the energy band this reduces to:

11. Benasque, August 31, 2012

12. - Coupling parameter for - Transmission coefficient Average transmission and variance - Variance of the transmission Analytical expressions for 1D continuous (i.e. not discrete) disordered media Excellent agreement between discrete Anderson model and formally exact analytical results for continuous system

13. Benasque, August 31, 2012 General case for any coupling: One can introduce an effective localization length: with where

14. Benasque, August 31, 2012