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  "body": "\n## Definition\n\nLet $q \\colon [0, \\infty) \\to \\R$ be continuous, let $h \\in \\R$, and consider the [[Schrödinger operator|89AA6XJY5K]] $-y'' + q y$ on $[0, \\infty)$ with the boundary condition $y'(0) = h y(0)$.\nFor $\\lambda \\in \\C$, let $\\varphi(x, \\lambda)$ be the solution of $-y'' + q y = \\lambda y$ with $\\varphi(0, \\lambda) = 1$ and $\\varphi'(0, \\lambda) = h$.\nLet $\\rho$ be the spectral function of the operator.\nThis is the nondecreasing function for which $f \\mapsto \\int_0^\\infty f(x) \\varphi(x, \\lambda) \\, dx$ extends to a unitary map from $L^2(0, \\infty)$ onto $L^2(\\R, d\\rho)$.\nLet $\\rho_0(\\lambda) = \\frac{2}{\\pi}\\sqrt{\\lambda}$ for $\\lambda \\ge 0$ and $\\rho_0(\\lambda) = 0$ for $\\lambda \u003c 0$; this is the spectral function for $q = 0$ and $h = 0$.\nSet\n$$F(x, t) = \\int_{-\\infty}^{\\infty} \\cos(\\sqrt{\\lambda}\\, x) \\cos(\\sqrt{\\lambda}\\, t) \\, d(\\rho - \\rho_0)(\\lambda),$$\nwhere the integral converges in a suitable regularised sense.\nThe Gelfand–Levitan equation is the linear integral equation for an unknown function $K(x, t)$:\n$$K(x, t) + F(x, t) + \\int_0^x K(x, s) F(s, t) \\, ds = 0, \\qquad 0 \\le t \u003c x.$$\nGelfand–Levitan theory is the method that recovers $q$ and $h$ from $\\rho$ by solving this equation and setting\n$$q(x) = 2 \\frac{d}{dx} K(x, x), \\qquad h = K(0, 0).$$\n\n## Remarks\n\nThe solution $K$ is the transformation kernel with $\\varphi(x, \\lambda) = \\cos(\\sqrt{\\lambda}\\, x) + \\int_0^x K(x, t) \\cos(\\sqrt{\\lambda}\\, t) \\, dt$.\nExample: for $q = 0$ and $h = 0$, one has $\\rho = \\rho_0$, so $F = 0$, the equation gives $K = 0$, and the formulas return $q = 0$ and $h = 0$.\nNon-example: an arbitrary nondecreasing function $\\rho$ need not be a spectral function of such an operator; Gelfand and Levitan gave conditions on $\\rho$ that characterise spectral functions.\nFor problems on a finite interval $[0, \\pi]$ the same method works with $\\rho$ a step function whose jumps sit at the eigenvalues.\n\n## Explanations\n\n### L1 human\n\nThe Gelfand–Levitan equation turns spectral data into a potential.\nOne measures how the given spectral function differs from the one of the free equation, solves a linear integral equation, and reads $q$ off the diagonal of the solution.\n",
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