Inverse spectral problem for the Laguerre differential operator (Q1270905)

In [SIAM Rev. 34, 118-119 (1992)] \textit{A. Zayed} asked whether there exists a perturbation of the Laguerre operator (with boundary condition at \(0\) that leads to the Friedrichs extension) which has a given equally spaced sequence of real numbers, bounded neither from above nor from below, as its spectrum. Prescribing certain normalization constants, the author shows by using the Gel'fand-Levitan theory that there is exactly one such permutation. Technical difficulties arise from the fact that this theory is usually phrased only for the situation where the principal coefficient and the weight function in the Sturm-Liouville operator are equal to one. An important ingredient of the analysis is an integral representation of the Laguerre functions by means of the Bessel function of order zero.