A note on eigenvalue asymptotics for Hill's equation
The nonlinear eigenvalue problem for a Hill's type equation is considered with respect to periodic and nonperiodic boundary conditions. Asymptotic approximations of the periodic and antiperiodic eigenvalues are given up to an order of \(o(n^{-1})\). The theorem of \textit{V. A. Amburzamyan} [Z. f. Physik 53, 690--695 (1929; JFM 55.0868.01)] concerning the reconstruction of the potential from the spectrum of a Sturm-Liouville problem with Neumann boundary conditions on a finite interval can be considered as the starting point for inverse spectral theory. Later, the theorem was extended to more general boundary conditions. In 1997, \textit{H. H. Chern} and \textit{C. L. Shen} [Inverse Probl. 13, No.~1, 15--18 (1997; Zbl 0866.34020)] presented a proof for a vectorial version of the theorem. A short proof of the Ambazumyan theorem for periodic and anti-periodic Sturm-Liouville problems is given here by applying the obtained asymptotic estimates.
scientific article; zbMATH DE number 5772529