Some inverse problems associated with Hill operator

2015/04/24 by Alp Arslan Kıraç, Alp Arslan Kirac, Kirac, Alp Arslan
Computer Science · Mathematics · #Matrix Theory and Algorithms #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #math.SP

paper · pdf · doi:10.48550/arxiv.1504.06547

arxiv created 2015/04/24 · arxiv updated 2015/04/27

Abstract

Let l_n be the length of the n-th instability interval of the Hill operator Ly=-y^′′+q(x)y. We obtain that if l_n=o(n^-2) then c_n=o(n^-2), where c_n are the Fourier coefficients of q. Using this inverse result, we prove: Let l_n=o(n^-2). If \(nπ)²: \textrmn even and n>n₀\ is a subset of the periodic spectrum of Hill operator then q=0 a.e., where n₀ is a positive large number such that l_n<ε n^-2 for all n>n₀(ε) with some ε>0. A similar result holds for the anti-periodic case.

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