Well-posed inverse spectral problems

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Abstract
It is known that if complete spectral data are provided, the potential function in a Sturm-Liouville operator is uniquely determined almost everywhere. If two such operators have spectra that differ in a finite number of eigenvalues, then the corresponding potential functions will no longer be the same. However, as is demonstrated when the nonidentical eigenvalues are almost equal, then the corresponding potential functions will also be nearly equal almost everywhere. Furthermore, if an operator and its spectrum are given and the potential is presumably known and if a second operator is defined in such a way such that its eigenvalues agree with the eigenvalues of the first operator except for a finite set, then the potential corresponding to the second operator can be explicitly found by solving a set of nonlinear ordinary differential equations. Lastly, it is shown that the procedures discussed here and the Gelfand-Levitan procedures are significantly different, and in fact that the Gelfand-Levitan procedure is almost certainly not well posed.

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