Inverse spectral problem

Remarks

The direct problem computes the eigenvalues from $q$; the inverse problem goes the other way. Ambartsumian showed in 1929 that for continuous $q$ on $[0, \pi]$ with the conditions $y'(0) = y'(\pi) = 0$, if the eigenvalues are exactly $0, 1, 4, 9, \ldots$, then $q = 0$. In general one spectrum does not determine $q$: for example, $q(x)$ and $q(\pi - x)$ on $[0, \pi]$ with Dirichlet conditions at both ends have the same eigenvalues. Göran Borg proved that two spectra, for two different boundary conditions at one end and the same condition at the other, determine $q$. On a half-line, the spectral measure, or equivalently the Weyl–Titchmarsh function, determines $q$; the Gelfand–Levitan method reconstructs it.