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