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Zbl 0753.35057
Gerard, C.; Martinez, A.; Sjöstrand, J.
A mathematical approach to the effective Hamiltonian in perturbed periodic problems. (English)
[J] Commun. Math. Phys. 142, No.2, 217-244 (1991). ISSN 0010-3616; ISSN 1432-0916/e

The authors 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.\par They study partial differential operators of the form $P=P(hy,y,D\sb y+A(hy))$ on $\bbfR\sp n$ (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. They 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 ${\cal E}(x,hD\sb x,h)$ acting on some Hilbert space depending on $\Gamma$.\par This is applied to the study of the Schrödinger operator when the electric potential is periodic, and to some quasiperiodic potentials with vanishing magnetic field.
[B.Helffer (Paris)]

MSC 2000:: *35P05 General spectral theory of PDE
81Q20 Semi-classical techniques in quantum theory
35S05 General theory of pseudodifferential operators
35J10 Schroedinger operator

Keywords: Schrödinger operator with periodic electric potential; semiclassical analysis

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