## Research Interests

My primary research area is mathematical physics.

More specifically, my recent research has been mainly focused on

*Study of adiabatic behavior in quantum mechanics and its applications. *The
theory of adiabatic evolution describes nonautonomous systems driven slowly by
external means and autonomous systems possessing two widely separated time
scales. The theory has proven to be fertile soil for interesting mathematical
physics and important applications. Adiabatic evolutions play a role in
celestial mechanics, plasma physics, in the Born-Oppenheimer theory of
molecules, in atom and ion traps, in the integer quantum Hall effect, quantum
pumps, in studies of the time dependent Ginzburg-Landau equation, and recently
in the quantum computation theory.

*Study of Anderson
localization on the lattice and in the continuum.* The addition of disorder
can have a profound effect on the spectral and dynamical properties of a self
adjoint differential operator. In general terms, the effect is that in certain
energy ranges the absolutely continuous spectrum of the Laplacian may be modified
to consist of a random dense set of eigenvalues associated with localized
eigenfunctions. Thus it affects various properties of the corresponding model:
time evolution (non--spreading of wave packets), conductivity (in response to electric
field), and Hall currents (in the presence of both magnetic and electric
field). This phenomenon, known as Anderson
localization (named after the Nobel prize laureate P. W. Anderson, who first
proposed it as a mechanism for localization in 1958) was initially discussed in
the context of the conduction properties of metals, but the mechanism is of
relevance in a variety of other situations. The behavior of the system depends
on the dimension **d**. In the one dimensional
case the Anderson Hamiltonian has a dense point spectrum for any non zero value
of the strength parameter.. In any dimension there are regimes (high disorder, extreme
energies) where the spectrum is dense point. The difficult open question in the
theory is the proof of the existence of the absolutely continuous spectral measures
in dimension higher than two. While it is widely believed by physicists that
such regime should exist, so far no rigorous result of this kind was proven.

**Last Update Aug 20, 2007**