The Rotation Number Approach to the Periodic Fuc caron ik Spectrum

In this paper, we study the Fu ik spectrum of the problem: (*) +(λ++q+(t))x++(λ−+q−(t))x−=0 with the 2π-periodic boundary condition, where q±(t) are 2π-periodic. After introducing a rotation number function ρ(λ+, λ−) for (*), we prove using the Hamiltonian structure and the positive homogeneity of (*) that for any positive integer n, the two boundary curves of the domain ρ−1(n/2) in the (λ+, λ−)-plane are Fu ik curves of (*). The result obtained in this paper shows that such a spectrum problem is much like that of the higher dimensional Fu ik spectrum with the Dirichlet condition. In particular, it remains open if the Fu ik spectrum of (*) is composed of only these curves