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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 \(P=P(hy,y,D_ y+A(hy))\) on \(\mathbb{R}^ n\) (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 \({\mathcal E}(x,hD_ x,h)\) 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.
Reviewer: B.Helffer (Paris)

MSC:

35P05 General topics in linear spectral theory for PDEs
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35S05 Pseudodifferential operators as generalizations of partial differential operators
35J10 Schrödinger operator, Schrödinger equation
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