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  "body": "\n## Summary\n\nGelfand–Levitan theory reconstructs the potential $q$ of a [[Sturm–Liouville|7WV6B2502K]] operator $-y'' + q y$ from its spectral measure.\nThe key step is a linear integral equation, the Gelfand–Levitan equation, whose kernel is built from the spectral data.\nIts solution $K(x, t)$ is a transformation kernel that turns cosines into the solutions of the perturbed equation, and $q(x) = 2 \\frac{d}{dx} K(x, x)$.\nIsrael Gelfand and Boris Levitan introduced it, and the related Marchenko equation is a basic tool of the inverse scattering transform.\n\n## Explanations\n\n### L1 human\n\nSuppose you know the notes a string can play, together with some extra information, and you want to find the varying stiffness of the string.\nGelfand–Levitan theory gives a recipe for this inverse problem.\nIt compares the given spectral data with those of a plain string, writes the difference as a kernel, and solves one linear integral equation.\nThe answer $q$ then appears from the diagonal values of the solution.\nWhen the spectral data are those of the plain string, the kernel is zero and the recipe returns $q = 0$.\n",
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