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.