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Triviality of Bloch and Bloch-Dirac bundles. (English) Zbl 1375.81102
Summary: In the framework of the theory of an electron in a periodic potential, we reconsider the longstanding problem of the existence of smooth and periodic quasi-Bloch functions, which is shown to be equivalent to the triviality of the Bloch bundle. By exploiting the time-reversal symmetry of the Hamiltonian and some bundle-theoretic methods, we show that the problem has a positive answer in any dimension \(d\leq 3\), thus generalizing a previous result by G. Nenciu [Commun. Math. Phys. 91, 81–85 (1983; Zbl 0545.47012)]. We provide a general formulation of the result, aiming at the application to the Dirac equation with a periodic potential and to piezoelectricity.

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
47N50 Applications of operator theory in the physical sciences
82D25 Statistical mechanical studies of crystals
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