On double eigenvalues of Hill's operator
The author considers Hill's equation \[ -y''+p(x)y=\lambda y \] with periodic boundary conditions \(y(x+2\pi)=y(x)\), where the potential p is in \(L^ 2(<0,\pi >)\), extended periodically to all of \({\mathbb{R}}\). Let \(\{\lambda_ k\}\), \(\lambda_ 0<\lambda_ 1\leq \lambda_ 2<\lambda_ 3\leq...\), be the periodic spectrum of p. \textit{S. P. Novikov} [Funct. Anal. Appl. 8(1974), 236-246 (1975; Zbl 0299.35017)] conjectured that the set of potentials p in \(L^ 2(<0,\pi >)\) with periodic spectrum having finitely many simple periodic eigenvalues is dense in the norm topology of \(L^ 2(<0,\pi >)\). This result has been proved by various authors, e.g. \textit{V. A. Marchenko} [Sturm Liouville Operators and Applications, Basel (1986; Zbl 0592.34011)], \textit{B. M. Levitan} [Math. USSR-Izv. 20, 55-87 (1983; Zbl 0517.34037)], \textit{J. Garnett} and \textit{E. Trubowitz} [Comment. Math. Helv. 62, 18-37 (1987; Zbl 0649.34034)]. All of them use the inverse spectral theory. In this paper a new proof is given without using the inverse spectral theory.
scientific article; zbMATH DE number 4142848