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An overview of periodic elliptic operators. (English) Zbl 1346.35170
This review article surveys the basic methods and results obtained in the theory of the multidimensional elliptic Schrödinger operators with the potential function in the form of a periodic function of the spatial coordinates. The covered topics include, in particular, the dispersion relation and bandgap spectra of these operators, their Green’s functions, quasiperiodic Bloch wave functions and localized wave packets in the form of Wannier functions, as well as analytical properties of Bloch and Fermi manifolds. The analysis described in this article is applicable not only to the standard Schrödinger operators with the spatially periodic potentials, but also to a wide variety of elliptic equations with periodic potentials, and to equations of that type on graphs. Physical applications which give rise to such equations, including photonic crystals, are considered in a brief form. All the analysis is presented only for linear equations.

MSC:
35Q40 PDEs in connection with quantum mechanics
35J10 Schrödinger operator, Schrödinger equation
35J15 Second-order elliptic equations
47A53 (Semi-) Fredholm operators; index theories
47F05 General theory of partial differential operators
58J05 Elliptic equations on manifolds, general theory
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