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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 $P=P(hy,y,D\sb y+A(hy))$ 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 $\Gamma$. 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.

35P05General topics in linear spectral theory of PDE
81Q20Semi-classical techniques in quantum theory, including WKB and Maslov methods
35S05General theory of pseudodifferential operators
35J10Schrödinger operator
Full Text: DOI
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