An inverse spectral theory of Gel'fand-Levitan-type for higher order differential operators (Q1068268)

An inverse spectral theory is presented for certain linear ordinary differential operators of arbitrary even order n which generalizes the Gel'fand-Levitan theory for Sturm-Liouville operators. It is proved that the coefficients in these operators are uniquely determined by n-1 distinct spectral matrices. Our method of proof makes use of a transformation due to M. K. Fage which generalizes the Povzner-Levitan transformations for Sturm-Liouville operators. The proof indicated in the paper is valid for operators of the form \(L=(- 1)\frac{^{n/2}d^ n}{dx^ n}q(x),\) but not for the more general operators indicated in the paper [see the erratum, ibid. 9, 263 (1985)]. Following a suggestion by J. R. McLaughlin, the author has proved that the coefficient q(x) is uniquely determined by the specification of one spectral matrix; it is not necessary to specify n-1 spectral matrices as is stated in the paper. This proof is contained in a paper ''An inverse scattering formalism for higher-order differential operators'' soon to be published in the J. Math. Anal. Appl.