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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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[1] [Ba] Balazard-Konlein, A.: Calcul fonctionnel pour des opérateursh-admissibles à symbole opérateur et applications. Thèse 3ème cycle, Nantes 1985
[2] [Be] Beals, R.: Characterization of pseudodifferential operators and applications. Duke Math. J.44, (1), 45–57 (1977) · Zbl 0353.35088 · doi:10.1215/S0012-7094-77-04402-7
[3] [Bu] Buslaev, V. S.: Semiclassical approximation for equations with periodic coefficients. Russ. Math. Surv.42 (6), 97–125 (1987) · Zbl 0698.35130 · doi:10.1070/RM1987v042n06ABEH001502
[4] [Gu-Ra-Tr] Guillot, J. C., Ralston, J., Trubowitz, E.: Semi-classical methods in solid state physics. Commun. Math. Phys.116, 401–415 (1988) · Zbl 0672.35014 · doi:10.1007/BF01229201
[5] [Ha] Harrell, E. M.: The band structure of a one dimensional periodic system in the scaling limit. Ann. Phys.119, 351–369 (1979) · Zbl 0412.34013 · doi:10.1016/0003-4916(79)90191-X
[6] [He-Sj] Helffer, B., Sjöstrand, J.: [1] On diamagnetism and de Haas-Van Alphen effect. Ann. I.H.P. (physique théorique)52 (4), 303–375 (1990)
[7] Analyse semi-classique pour l’équation de Harper. Bull. S.M.F., Mémoire no 34, T. 116, Fasc. 4 (1988) · Zbl 0714.34130
[8] [He-Sj] Helffer, B., Sjöstrand, J.: [3] Semiclassical analysis for Harpers equation III–Cantor structure of the spectrum. Bull. S.M.F., Mémoire (to appear) · Zbl 0725.34099
[9] [Hö] Hörmander, L.: The Weyl calculus of pseudodifferential operators. Commun. Pure App. Math.32, 359–443 (1979) · Zbl 0396.47029 · doi:10.1002/cpa.3160320304
[10] [Ne] Nenciu, G.: Dynamics of band electrons in electric and magnetic fields: rigorous justification of the effective Hamiltonians. Rev. Mod. Phys.63, 91–127 (1991) · doi:10.1103/RevModPhys.63.91
[11] [Ou] Outassourt, A.: Analyse semi-classique pour des opérateurs de Schrödinger avec potential périodique. J. Funct. Anal.72 (1), (1987)
[12] [Re-Si] Reed, M., Simon, B.: Methods of modern mathematical physics, Tome IV. New York: Academic Press 1975
[13] [Si] Simon, B.: Semiclassical analysis of low lying eigenvalues III–Width of the ground state band in strongly coupled solids. Ann. Phys.158, 415–420 (1984) · Zbl 0596.35028 · doi:10.1016/0003-4916(84)90125-8
[14] [Sk] Skriganov, M. M.: Geometric and arithmetic methods in the spectral theory of multidimensional periodic operators. Proceedings of the Steklov Institute of Mathematics, no2 (1987) · Zbl 0615.47004
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