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A mathematical approach to the effective Hamiltonian in perturbed periodic problems. (English) Zbl 0753.35057

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.

They study partial differential operators of the form $P=P\left(hy,y,{D}_{y}+A\left(hy\right)\right)$ on ${ℝ}^{n}$ (when $h>0$ is small enough), where $P\left(x,y,\eta \right)$ is elliptic, periodic in $y$ with respect to some lattice ${\Gamma }$, and admits smooth bounded coefficients in $\left(x,y\right)$. $A\left(x\right)$ 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 $ℰ\left(x,h{D}_{x},h\right)$ acting on some Hilbert space depending on ${\Gamma }$.

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.

##### MSC:
 35P05 General topics in linear spectral theory of PDE 81Q20 Semi-classical techniques in quantum theory, including WKB and Maslov methods 35S05 General theory of pseudodifferential operators 35J10 Schrödinger operator
##### References:
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