2008/08/05 by Andrey Badanin, Badanin, Andrey, Evgeny Korotyaev +1
Computer Science · Mathematics · Physics and Astronomy · #34L20 #34L40 #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #math-ph #math.MP #msc:34L20 #msc:34L40
paper · pdf · doi:10.48550/arxiv.0808.0588
26 pages
arxiv created 2008/08/05 · openalex publication_date 2008/08/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the operator H=d4dt4+ddtpddt+q with 1-periodic coefficients on the real line. The spectrum of H is absolutely continuous and consists of intervals separated by gaps. We describe the spectrum of this operator in terms of the Lyapunov function, which is analytic on a two-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the scalar case. We describe the spectrum of H in terms of periodic, antiperiodic eigenvalues, and so-called resonances. We prove that 1) the spectrum of H at high energy has multiplicity two, 2) the asymptotics of the periodic, antiperiodic eigenvalues and of the resonances are determined at high energy, 3) for some specific p the spectrum of H has an infinite number of gaps, 4) the spectrum of H has small spectral band (near the beginner of the spectrum) with multiplicity 4 and its asymptotics are determined as p→ 0, q=0.