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  "body": "\n## Problem\n\nLet $q$ be a real potential on an interval, and consider the [[Schrödinger operator|89AA6XJY5K]] $-y'' + q y$ with given boundary conditions.\nDetermine which spectral data of this operator determine $q$ uniquely, and reconstruct $q$ from such data.\n\n## Remarks\n\nThe direct problem computes the eigenvalues from $q$; the inverse problem goes the other way.\nAmbartsumian 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 $q = 0$.\nIn 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.\nGöran Borg proved that two spectra, for two different boundary conditions at one end and the same condition at the other, determine $q$.\nOn a half-line, the spectral measure, or equivalently the Weyl–Titchmarsh function, determines $q$; the [[Gelfand–Levitan|R4MRNP9EQT]] method reconstructs it.\n\n## Explanations\n\n### L1 human\n\nIf you hear all the notes a string can play, can you tell how its thickness varies along its length?\nThis is the inverse spectral problem for a string.\nThe answer is no from the notes alone, because a string and its mirror image sound the same, but yes with a little more information.\n",
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      "first_stated": "1929",
      "posed_by": "Viktor Ambartsumian"
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