The semiclassical electron in a magnetic field and lattice. Some problems of low dimensional “periodic” topology.(English)Zbl 0853.57014
The general theory of Hamiltonian systems and foliations on surfaces is adopted to the semiclassical (or adiabatic) motion of the quantum electron in the periodic lattice and weak magnetic field. A number of problems is formulated, including the fundamental question:
Which kind of observable quantities in the conductivity of normal metals may correspond to the integer-valued topological invariants of generic open orbits in the external homogeneous weak magnetic field.
Which kind of observable quantities in the conductivity of normal metals may correspond to the integer-valued topological invariants of generic open orbits in the external homogeneous weak magnetic field.
Reviewer: P.Garbaczewski (Wrocław)
MSC:
| 57M50 | General geometric structures on low-dimensional manifolds |
| 37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
Cite
References:
| [1] | [D]I. Dynnikov, Proof of Novikov’s Conjecture on the semiclassical motion of electron, Math. Zametki 53:5 (1993), 57–68. ·Zbl 0808.58014 |
| [2] | [GN]P.G. Grinevich, S.P. Novikov, String equation – 2. Physical solution, Algebra and Analysis (1994), (dedicated to the 60th birthday of L.D. Faddeev). ·Zbl 0836.35142 |
| [3] | [A]A.A. Abrikosov, Introduction to the Theory of Metals, Moscow, Nauka (1987). |
| [4] | [N1]S.P. Novikov, The Hamiltonian formalism and a multivalued analog of Morse theory, Uspekhi Math. Nauk (Russian Math Surveys) 37:5 (227) (1982), 3–49. |
| [5] | [N2]S.P. Novikov, Critical points and level surfaces of multivalued functions. Proceedings of the Steklov Institute of Mathematics, 1986, AMS, iss 1, 223–232. ·Zbl 0588.58006 |
| [6] | [N3]S.P. Novikov, Quasiperiodic structures in topology. In the proceedings of the Conference ”Topological methods in Modern Mathematics”, Stonybrook University, June 1991 (dedicated to the 60th birthday of John Milnor). Stonybrook, 1993. |
| [7] | [N4]S.P. Novikov, Quantization of the finite gap potentials and string equation, Functional Analysis and its Applications 24:4 (1990), 196–206. |
| [8] | [N5]S.P. Novikov, Two dimensional Schroedinger Operator in the periodic fields. Current Problems in Mathematics, VINITI, 1983, v 23, 3–22. (Translated by AMS in the January 1985). |
| [9] | [N6]S.P. Novikov, Bloch functions in a magnetic field and vector bundles. Typical dispersion relations and their quantum numbers, Doklady AN SSSR (Russian Math Dokl) 257:3 (1981), 538–543. |
| [10] | [Z]A.V. Zorich, Novikov’s problem on the semiclassical motion of electron in the homogeneous magnetic field. Uspekhi Math Nauk (RMS) 39:5 (1984), 235–236. ·Zbl 0900.58031 |
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