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 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.
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Boumenir, A., Tuan, V. The Gelfand-Levitan Theory Revisited. J Fourier Anal Appl 12, 257–267 (2006). https://doi.org/10.1007/s00041-005-5075-9
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DOI: https://doi.org/10.1007/s00041-005-5075-9