The Gelfand-Levitan Theory Revisited

Gelfand and Levitan in their celebrated article in 1951, and later Gasymov and Levitan in 1964 have shown that a monotone increasing function is a spectral function of a singular Sturm-Liouville problem on a half-line in the limit point case at infinity if and only if it satisfies an existence and a smoothness condition. In this article, a closer look at the original statement reveals that the existence condition in fact follows from the smoothness condition which simplifies significantly the statement of the Gelfand-Levitan theory. We also provide two sufficient and verifiable conditions for a nondecreasing function to be the spectral function of a singular Sturm Liouville operator.