Abstract
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 formP=P(hy, y, D y +A(hy)) onR n (whenh>0 is small enough), whereP(x, y, η) is elliptic, periodic iny with respect to some lattice Γ, and admits smooth bounded coefficients in (x, y).A(x) is a magnetic potential with bounded derivatives. We show that the spectral study ofP near any fixed energy level can be reduced to the study of a finite system ofh-pseudodifferential operatorsE(x, hD x , h), acting on some Hilbert space depending on Γ. We then apply it to the study of the Schrödinger operator when the electric potential is periodic, and to some quasiperiodic potentials with vanishing magnetic field.
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Gerard, C., Martinez, A. & Sjöstrand, J. A mathematical approach to the effective Hamiltonian in perturbed periodic problems. Commun.Math. Phys. 142, 217–244 (1991). https://doi.org/10.1007/BF02102061
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DOI: https://doi.org/10.1007/BF02102061