Erratum to: Floquet's theorem and stability of periodic solitary waves (Q607514)

This paper is concerned with the spectrum of the Hill operator \[ L(y) = - y^{\prime \prime} + Q(x) y \] in \(L_{per}^2 [0, \pi]\), where the potential \(Q(x)\) is a \(C^2\) \(\pi\)-periodic function. It is known that the spectrum of \(L\) is an unbounded sequence of real numbers \(\lambda_0 < \lambda_1 \leq \lambda_2 < \lambda_3 \leq \lambda_4 \cdots < \lambda_{2n-1} \leq \lambda_{2n} \cdots\). It is also known that (see Theorem 2.1 and 2.14 in [\textit{W. Magnus} and \textit{S. Winkler}, Hill's Equation. New York-London-Sydney: Interscience Publishers, a division of John Wiley \& Sons (1966; Zbl 0158.09604)]) if \(p(x)\) is an eigenfunction of \(L\) associated to the eigenvalue \(\lambda\), and if \(p(x)\) has \(2n\) zeros in \([0, \pi)\), then \(\lambda\) is either \(\lambda_{2n-1}\) or \(\lambda_{2n}\). By means of a new version for the second part of Floquet's theorem, the author improves this result. He shows that \(p(x)\) contains all the information needed to characterize \(\lambda\), despite of \(\lambda\) being simple or not. In case that \(\lambda\) is simple, it holds that \(\lambda_{2n-1}\) or \(\lambda_{2n}\). Applications are given to th problem of nonlinear stability of a class of periodic problems. This is a corrected republication of the article originally published in [the author, ibid. 21, No.~3, 555--565 (2009; Zbl 1235.34226)].