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.

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)

##### MSC:

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 |