+ q with periodic coefficients p, q on the real line is considered. The spectrum of H is absolutely continuous and consists of intervals separated by gaps. The following statements are proved: 1) the endpoints of gaps are periodic or antiperiodic eigenvalues or branch points of the Lyapunov function, and moreover, their asymptotic behavior at high energy is found; 2) the spectrum of H at high energy has multiplicity two; 3) if p belongs to a certain class, then for any q the spectrum of H has infinitely many gaps, and all branch points of the Lyapunov function, except for a finite number of them, are real and negative; 4) if q = 0 and p → 0, then at the beginning of the spectrum there is a small spectral band of multiplicity 4, and its asymptotic behavior is found; the remaining spectrum has multiplicity 2. §1. Introduction and main results
Consider a selfadjoint operator Hand has real 1-periodic coefficients p and q such that p, p , q ∈ L 1 (0, 1). Henceforth we use the notationIt is known that the spectrum of the operator H is absolutely continuous and consists of nondegenerate intervals (see [8, Theorem XIII.7.64]). We introduce the fundamental solutions ϕ j (t, λ), j ∈ N 3 = {0, 1, 2, 3}, of the equation, where δ jk is the Kronecker symbol. We define a 4 × 4 monodromy matrix M and the characteristic polynomial D by the relations
It is known that
We consider 2p 4 order differential operator on the real line with a periodic coefficients. The spectrum of this operator is absolutely continuous and is a union of spectral bands separated by gaps. We define the Lyapunov function, which is analytic on a p-sheeted Riemann surface. The Lyapunov function has real or complex branch points. We prove the following results: (1) The spectrum at high energy has multiplicity two. (2) Endpoints of all gaps are periodic (or anti-periodic) eigenvalues or real branch points. (3) The spectrum of operator has an infinite number of open gaps and there exists only a finite number of non-real branch points for some specific coefficients (the generic case). (4) The asymptotics of the periodic, anti-periodic spectrum and branch points are determined at high energy.
We consider resonances for fourth order differential operators on the half-line with compactly supported coefficients. We determine asymptotics of a counting function of resonances in complex discs at large radius, describe the forbidden domain for resonances and obtain trace formulas in terms of resonances. We apply these results to the Euler-Bernoulli operator on the half-line. The coefficients of this operator are positive and constants outside a finite interval. We show that this operator does not have any eigenvalues and resonances iff its coefficients are constants on the whole half-line.
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