BKW construction for the Dirac operator -: Case of a periodic potential

In the first part of this work, we compute explicitly the principal symbol of a WKB solution of the Dirac operator along a planar Agmon geodesic starting from a non-degenerate ponctual well. In the second part, we apply this result to study the band spectrum of a periodic Dirac operator near the minimum of the potential, we construct a class of potentials where the multiplicity of the Floquet eigenvalues is 1 and not 2 (eigenvalues of a Dirac operator with zero magnetic field have even multiplicity). (C) Elsevier, Paris.