Effects of Disorder on Quantum Spectra and Dynamics

Michael Aizenman


From the earliest successes of quantum mechanics, with the absorption and emission spectrum of hydrogen, to some of the latest advances in condensed matter physics, an important role of the theory has been in explaining the energy spectra, and the nature of the corresponding "eigenmodes" of physical systems.  In cases of interest for condensed matter physics, these
correspond to vibrations of extended systems of particles, often with a periodic order.  In periodically ordered systems the vibrational modes are extended, however the nature of the spectrum and of the eigenmodes can be drastically affected by disorder.  In particular, the vibrational modes may become localized.  The phenomenon, known as Anderson localization, 
has strong effects on conduction, and plays also an important role in the Quantum Hall Effect.  The talk will focus on the mathematical analysis of the nature of the spectra and dynamics in systems with extended, though  possibly weak, disorder.  More specifically, we shall discuss recent advances
in the study of the spectral and dynamical properties of Schroedinger operators with random potentials.  While the problem is one of functional analysis, a key role is played by techniques from mathematical statistical mechanics.