Gianluca Panati

Learn More
We reconsider the longstanding problem of an electron moving in a crystal under the influence of weak external electromagnetic fields. More precisely we analyze the dynamics generated by the Schrödinger operator H = 1 2 (−i∇x −A(εx)) 2 + V (x) + φ(εx), where V is a lattice periodic potential and A and φ are external potentials which vary slowly on the scale(More)
The exponential localization of Wannier functions in two or three dimensions is proven for all insulators that display time-reversal symmetry, settling a long-standing conjecture. Our proof relies on the equivalence between the existence of analytic quasi-Bloch functions and the nullity of the Chern numbers (or of the Hall current) for the system under(More)
In the framework of the theory of an electron in a periodic potential, we reconsider the longstanding problem of the existence of smooth and periodic quasi-Bloch functions, which is shown to be equivalent to the triviality of the Bloch bundle. By exploiting the time-reversal symmetry of the Hamiltonian and some bundle-theoretic methods, we show that the(More)
A systematic perturbation scheme is developed for approximate solutions to the time-dependent Schrödinger equation with a space-adiabatic Hamiltonian. For a particular isolated energy band, the basic approach is to separate kinematics from dynamics. The kinematics is defined through a subspace of the full Hilbert space for which transitions to other band(More)
Let VΓ be a lattice periodic potential and A and φ external electromagnetic potentials which vary slowly on the scale set by the lattice spacing. It is shown that the Wigner function of a solution of the Schrödinger equation with Hamiltonian operator H = 1 2 (−i∇x − A(εx)) 2 + VΓ(x) + φ(εx) propagates along the flow of the semiclassical model of solid(More)
A rigorous nonperturbative adiabatic approximation of the evolution operator in the many-body physics of degenerate systems is derived. This approximation is used to solve the long-standing problem of the choice of the initial states of H(0) leading to eigenstates of H(0) + V for degenerate systems. These initial states are eigenstates of P(0)VP(0), where(More)
The flux-across-surfaces conjecture represents a corner stone in quantum scattering theory because it is the key-assumption needed to prove the usual relation between differential cross section and scattering amplitude. We improve a recent result [TDMB] by proving the conjecture also in presence of zero-energy resonances or eigenvalues, both in point and(More)
As a simple model for piezoelectricity we consider a gas of infinitely many non-interacting electrons subject to a slowly timedependent periodic potential. We show that in the adiabatic limit the macroscopic current is determined by the geometry of the Bloch bundle. As a consequence we obtain the King-Smith and Vanderbilt formula up to errors smaller than(More)