Multiplicities of the eigenvalues of periodic Dirac operators

Let us consider the Dirac operator L = i J d/dx + U, J = ((1)(0) (0)(-1)), U = ((a cos 2pix) (0) (0) (a cos 2pix)). where a not equal 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)) is an element of H-1(I). Then sigma(L-bc) = -sigma(L-bc), and all lambda is an element of sigma(Per)+(L(U)) are of multiplicity 2, while lambda is an element of sigma(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 nu = +/-p' + p(2) (Section 3). It helps to get analogues of Grigis' results (Ann. Sci. Ecole 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). (C) 2004 Elsevier Inc. All rights reserved.