The Journal of Fourier Analysis and Applications Volume 12, Issue 3, 2006 The Gelfand-Levitan Theory Revisited Amin Boumenir and Vu Kim Tuan Communicated by Frank Natterer ABSTRACT. 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 simpli- fies 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. 1. Introduction We are concerned with the solvability conditions of the inverse spectral problem for the singular Sturm-Liouville (S-L) operator, in the limit-point case at x =∞  L(y) := -y ′′ (x, λ) + q(x)y(x,λ) = λy(x, λ), x ∈[0, ∞) y ′ (0, λ) - hy(0, λ) = 0 (1.1) where q ∈ L 1,loc [0, ∞), q(x) and h are real. Recall that in the celebrated 1951 article by Gelfand and Levitan [5], the necessary and sufficient conditions were stated separately. To close the gap, in 1953, [7], M. G. Krein announced two necessary and sufficient conditions for ρ to be a spectral function which he then revised by adding a third condition in 1957, [8]. Few years later in 1964, Gasymov and Levitan closed the gap of the original 1951 result by obtaining two, necessary and sufficient, conditions for the solvability of the inverse spectral problem. To briefly state these conditions, denote by σ (λ) := ρ(λ) - 2 π √ λ + where λ + = max (0, λ) and F c (f ) (λ) =  ∞ 0 f(x) cos ( x √ λ ) dx the classical Fourier cosine transform. Math Subject Classifications. 34A55. Keywords and Phrases. Gelfand and Levitan theory, inverse spectral problem. © 2006 Birkhäuser Boston. All rights reserved ISSN 1069-5869 DOI: 10.1007/s00041-005-5075-9