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Exact energy bands and Fermi surfaces of separable abelian potentials. (English) Zbl 1001.81017

Summary: We present a general theory for multidimensional Schrödinger equations with separable Abelian potentials with an arbitrary number of gaps in the spectrum. In particular, we derive general equations which allow one to express the energy and the wavevectors in the Brillouin zone as a function of the spectral parameters. By using the solutions of these equations, we show how to construct the energy bands and the Fermi surfaces in the first Brillouin zone of the reciprocal lattice. As illustrative examples we consider the case of two-dimensional separable potentials with one, two and three gaps in the spectrum. The method can be applied to crystals with a cubic or a rectangular parallelogrammatic Wigner-Seitz cell in arbitrary dimensions. The possibility to generalize the theory to other crystal symmetries is also briefly discussed.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
82D37 Statistical mechanics of semiconductors
81V70 Many-body theory; quantum Hall effect
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