Some inverse results for Hill's equation
type: Item
instance of: scholarly article
title: Some inverse results for Hill's equation
zbMATH Open document ID: 1024.34005
DOI: 10.1016/S0022-247X(02)00454-7
author: Haskiz Coşkun
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-02-11T00:00:00Z
review text: The author deals with an inverse problem for the Hill equation \[ y''(t)+ (\lambda- q(t)) y(t)= 0\tag{\(*\)} \] on \(\mathbb{R}\), where \(\lambda\) is a real parameter and \(q\) is a real-valued, locally integrable, periodic function of period \(\pi\). Denote by \(l_n\) the length of the \(n\)th instability interval of this equation. Let \(\{a^k_n\}^\infty_{n=0}\) and \(\{b^k_n\}^\infty_{n=1}\) be the cosine and sine Fourier coefficients of \(q^{(k)}\), \(k= 0,1,2,\dots\), respectively. The author proves \hskip 15mm (1) If \(l_m={\mathcal O}(n^{-1})\), then \(a_n, b_n={\mathcal O}(n^{-1})\) as \(n\to\infty\). \hskip 15mm (2) If \(l_n={\mathcal O}(n^{-(k+2)})\), then \(a^k_n, b^k_n={\mathcal O}(n^{-2})\) as \(n\to\infty\) and hence \(q^{(k)}\) is absolutely continuous. In the proof of these results, the following technics are employed. Thanks to the theorem by \textit{H. Hochstadt} [Arch. Ration. Mech. Anal. 19, 353--362 (1965; Zbl 0128.31201)], the problem reduces to the study of the Dirichlet eigenvalues of \((*)\) on shifted intervals \([\tau,\tau+ \pi]\), \(\tau\in [0,\pi)\). Using the Prüfer transformation, the author obtains a leading asymptotic expansion of such Dirichlet eigenvalues and apply that to prove (1). The proof of (2) is based on the author's and \textit{B. J. Harris'} earlier result [Proc. R. Soc. Edinb., Sect. A, Math. 130, 991--998 (2000; Zbl 0977.34077)] on the higher-order asymptotic expansion, which was demonstrated by analyzing the associated Riccati equation.
reviewed by: Kazushi Yoshitomi
Mathematics Subject Classification ID: 34A55
Mathematics Subject Classification ID: 34L20
zbMATH DE Number: 1866779
zbMATH Keywords: Hill's equation
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: asymptotics
zbMATH Keywords: associated Riccati equation
MaRDI profile type: Publication
cites work: Estimates for the periodic and semi-periodic eigenvalues of Hill's equation
cites work: On the spectrum of a second-order periodic differential equation
cites work: Q4774804
cites work: Gaps and bands of one dimensional periodic Schrödinger operators
cites work: Asymptotics of eigenvalues for regular Sturm-Liouville problems
cites work: On the determination of a Hill's equation from its spectrum
cites work: Estimates on the Stability Intervals for Hill's Equation
cites work: Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points
cites work: The inverse problem for periodic potentials
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description: scientific article; zbMATH DE number 1866779