Multiplicities of the Eigenvalues of Periodic Dirac Operators

Let us consider the Dirac operator L = iJ d dx + U, J = ( 1 0 0 −1 ) , U = ( 0 a cos 2πx a cos 2πx 0 ) , where a 6= 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 = ( f1 f2 ) ∈ H(I). Then σ(Lbc) = −σ(Lbc), and all λ ∈ σPer+(L(U)) are of multiplicity 2, while λ ∈ σPer−(L(U)) are simple (Thm 15). This is an analogue of E. L. Ince’s statement for Mathieu-Hill operator. Links between spectra of Dirac and Hill operators lead to detailed information about spectra of Hill operators with potentials of the Ricatti form v = ±p′ + p (Section 3). It helps to get analogues of Grigis’ results [8] on zones of instability of Hill operators with polynomial potentials and their asymptotics for the case of Dirac operators as well (Section 4.2). keywords: Dirac operator, periodic potential, Hill operator, eigenvalue multiplicity, zones of instability.