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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.

##### MSC:
 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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