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Microlocal analysis for the periodic magnetic Schrödinger equation and related questions. (English) Zbl 0761.35090
Microlocal analysis and applications, Lect. 2nd Sess. CIME, Montecatini Terme/Italy 1989, Lect. Notes Math. 1495, 237-332 (1991).
[For the entire collection see Zbl 0747.00025.]
This course is devoted to a presentation of the microlocal methods applied to problems in solid state physics. The following material is presented: Floquet theory, stability of the gap for the Schrödinger equation with magnetic field, magnetic matrices, density of states, Harper’s equation, de Haas-van Alphen effect.
The interesting fact is the appearance in a lot of different contexts of an effective Hamiltonian which can be considered as a pseudo-differential operator with a small parameter.
These lectures are mainly based on joint work with B. Helffer but contain also original results of the author.
Reviewer: B.Helffer (Paris)

35Q40 PDEs in connection with quantum mechanics
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35-03 History of partial differential equations
Zbl 0747.00025