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Abstract

The matrix-valued function M is entire. An eigenvalue of M(λ) is called a multiplier. It is a root of the algebraic equation D(τ, λ) = 0, where D(τ, λ) ≡ det(M(λ) − τI 4 ), τ, λ ∈ C. Let D ± (λ) = (1/4)D(±1, λ). The zeros of D + (λ) (or D − (λ)) are the eigenvalues of the periodic (anti-periodic) problem for the equation y + Vy = λy. Denote by λ + 0 , λ ± 2n , n = 1, 2, . . . ,If for some λ ∈ C (or λ ∈ R) τ(λ) is a multiplier of multiplicity d 1, then τ −1 (λ) (or τ(λ))is a multiplier of multiplicity d. Moreover, each M(λ), λ ∈ C, has exactly four multipliersis a simple multiplier and |τ(λ)| = 1, then τ (λ) = 0.The spectral problems for the fourth-order periodic operator were the subject of many authors (see [2,3,19,24,25,27,28,31]). Firstly, we mention the papers of Papanicolaou [24, 25] devoted to the Euler-Bernoulli equation (ay ) = λby with the periodic functions a, b. For this case, he defines the Lyapunov function and obtains some properties of this function. In particular, it is proved that the Lyapunov function is analytic on some two-sheeted Riemann surface. It is important that for this case he proved that all branch points of the Lyapunov function are real and 0. Note that in our case we have the example, Proposition 1.7, where the Lyapunov function has real and nonreal branch points. This the main difference between our Lyapunov function and his one. Moreover, Papanicolaou proved that if all the gaps are closed and the Lyapunov at University of Sydney on March 15, 2015 http://imrn.oxfordjournals.org/ Downloaded from Spectral Asymptotics for Periodic Fourth-Order Operators 2777(1.8)We have ρ(λ) = ρ 0 (λ)(1 + o(1)) as |λ| → ∞, λ ∈ D 1 (see Lemma 5.1). Then we define the analytic function ρ(λ), λ ∈ D r , for some large r > 0, by the condition ρ(λ) = ρ 0 (λ)(1 + o(1)) as |λ| → ∞, λ ∈ D r , where ρ 0 (λ) = (cos z − cosh z)/2.The function ρ is real on R, then r is a root of ρ if and only if r is a root of ρ.By Lemma 5.1, for large integer N, the function ρ(λ) has exactly 2N + 1 roots, counted with multiplicity, in the disk {λ : |λ| < 4(π(N + 1/2)) 4 } and for each n > N, exactly two roots, counted with multiplicity, in the domain {λ : |λ 1/4 − π(1 + i)n| < π/4}. There are no