On the global quasimomentum in solid state physics. (English) Zbl 0996.81124

Bytsenko, A. A. (ed.) et al., Mathematical methods in physics. Proceedings of the 1999 Londrina Winter School, Londrina, Brazil, August 17-26, 1999. Singapore: World Scientific. 98-141 (2000).
Summary: The one-dimensional Schrödinger operator with a periodic potential (Hill operator), that is the energy operator in crystal lattice, is considered. The analytical basis for its study is the Riemann surface of the global quasimomentum introduced in [N. E. Firsova, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. 51, 183-196 (1975; Zbl 0349.34020)] to solve the scattering problem in crystals with impurities [N. E. Firsova, Mat. Zametki 18, 831-843 (1975; Zbl 0321.34016); Mat. Sb., Nov. Ser. 130(172), No. 3(7), 349-385 (1986 Zbl 0616.34017)], using the quasimomentum representation constructed there. Now, this global quasimomentum proved to be the fundamental notion with a large sphere of applications which are outlined.
The main characteristics such as effective masses, Floquet-Bloch functions etc. are studied. In particular it is found that the sum of all effective masses absolutely converges and is equal to the physical mass of the particle in the one dimensional crystal lattice which can be interpreted as the regularized trace formula for the Hill operator [see N. E. Firsova, On effective mass of the one-dimensional Hill operator, Vestn. Leningr. Gosudarst. Univ., SEr. Fiz. Khim. 1979, No. 2, 13-18 (1979); Teor. Mat. Fiz. 37, No. 2, 281-288 (1978; Zbl 0401.34025)].
For the entire collection see [Zbl 0943.00079].


81V70 Many-body theory; quantum Hall effect
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
82D20 Statistical mechanics of solids