A MATHEMATICAL APPROACH TO THE EFFECTIVE HAMILTONIAN IN PERTURBED PERIODIC PROBLEMS

We describe a rigorous mathematical reduction of the spectral study for a class of periodic problems with perturbations which gives a justification of the method of effective Hamiltonians in solid state physics. We study the partial differential operators of the form P = P(hy, y, D(y) + A(hy)) on IR(n) (when h > 0 is small enough), where P(x, y, eta) is elliptic, periodic in y with respect to some lattice GAMMA, and admits smooth bounded coefficients in (x, y). A(x) is a magnetic potential with bounded derivatives. We show that the spectral study of p near any fixed energy level can be reduced to the study of a finite system of h-pseudodifferential operators E(x, hD(x), h), acting on some Hilbert space depending on GAMMA. We then apply it to the study of the Schrodinger operator when the electric potential is periodic, and to some quasiperiodic potentials with vanishing magnetic field.