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  "body": "\n## Summary\n\nThe Gelfand–Levitan–Marchenko equation is a linear [[integral equation|R3SG1SN7JQ]] that recovers a potential $u$ in the one-dimensional Schrödinger operator $-\\frac{d^2}{dx^2} + u(x)$ from its scattering data.\nThe data are the reflection coefficient, the bound state energies and their norming constants; they enter the equation through a single function $F$.\nThe solution $K(x, y)$ gives the potential by $u(x) = -2 \\frac{d}{dx} K(x, x)$.\nIt is the central step of the [[inverse scattering transform|6GY3X4Y8X5]], which solves the [[Korteweg–De Vries equation|KTT75QC72Z]] and other integrable equations.\n\n## Explanations\n\n### L1 human\n\nSend waves at an unknown bump in a potential and record what comes back: how much of each wave is reflected and which waves stay trapped.\nThe inverse problem asks for the bump from these echoes.\nThe Gelfand–Levitan–Marchenko equation turns this question into a linear integral equation, which is much easier than the nonlinear problem it replaces.\nWhen there is no reflection and one trapped state, the equation can be solved by hand and gives the potential $-2\\kappa^2 \\operatorname{sech}^2(\\kappa(x - x_0))$, the shape of a [[soliton|63D4AXYCT4]].\n",
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