Multiplicities of the eigenvalues of periodic Dirac operators

Abstract

Let us consider the Dirac operator L = iJ d dx + U , J = 1 0 0 - 1 , U = 0 a cos 2 π x a cos 2 π x 0 , where a ≠ 0 is real, on I = [ 0 , 1 ] with boundary conditions bc = Per + , i.e., F ( 1 ) = F ( 0 ) , and bc = Per - , i.e., F ( 1 ) = - F ( 0 ) , F = f 1 f 2 ∈ H 1 ( I ) . Then σ ( L bc ) = - σ ( L bc ) , and all λ ∈ σ Per + ( L ( U ) ) are of multiplicity 2, while λ ∈ σ Per - ( L ( U ) ) are simple (Theorem 15). This is an analogue of Ince's statement for Mathieu–Hill operator. Links between the spectra of Dirac and Hill operators lead to detailed information about the spectra of Hill operators with potentials of the Ricatti form v = ± p ′ + p 2 (Section 3). It helps to get analogues of Grigis’ results (Ann. Sci. École Norm. Sup. (4) 20 (1987) 641) on the zones of instability of Hill operators with polynomial potentials and their asymptotics for the case of Dirac operators as well (Section 4.2).