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  "body": "\n## Definition\n\nLet $u \\colon \\R \\to \\R$ be a potential with $\\int_{\\R} (1 + \\abs{x}) \\abs{u(x)} \\, dx \u003c \\infty$, and consider $-\\psi'' + u \\psi = k^2 \\psi$ on $\\R$.\nLet its scattering data be the reflection coefficient $r(k)$ for $k \\in \\R$, the bound state energies $-\\kappa_1^2, \\ldots, -\\kappa_N^2$ with $\\kappa_j \u003e 0$, and the norming constants $\\beta_1, \\ldots, \\beta_N \u003e 0$.\nPut\n$$F(x) = \\frac{1}{2\\pi} \\int_{-\\infty}^{\\infty} r(k) \\, e^{ikx} \\, dk + \\sum_{j=1}^{N} \\beta_j e^{-\\kappa_j x}.$$\nThe *Gelfand–Levitan–Marchenko equation* is the equation for an unknown function $K(x, y)$, $y \\ge x$:\n$$K(x, y) + F(x + y) + \\int_x^\\infty K(x, z) \\, F(z + y) \\, dz = 0, \\quad y \u003e x.$$\nFor each fixed $x$ it is a linear integral equation of the second kind on $(x, \\infty)$, and the potential is recovered as $u(x) = -2 \\frac{d}{dx} K(x, x)$.\n\n## Remarks\n\nExample: the reflectionless data $r = 0$, $N = 1$, give $F(x) = \\beta e^{-\\kappa x}$.\nTry $K(x, y) = L(x) e^{-\\kappa y}$; the equation becomes $L(x) + \\beta e^{-\\kappa x} + L(x) \\frac{\\beta}{2\\kappa} e^{-2\\kappa x} = 0$.\nWith $w(x) = \\frac{\\beta}{2\\kappa} e^{-2\\kappa x}$ this gives $K(x, x) = -\\frac{2\\kappa w}{1 + w}$, and since $w' = -2\\kappa w$,\n$$u(x) = -2 \\frac{d}{dx} K(x, x) = -\\frac{8 \\kappa^2 w}{(1 + w)^2} = -2\\kappa^2 \\operatorname{sech}^2\\!\\big(\\kappa (x - x_0)\\big),$$\nwhere $e^{2\\kappa x_0} = \\frac{\\beta}{2\\kappa}$.\nThis potential has the single bound state energy $-\\kappa^2$ and reflects nothing.\n\nNon-example: the equation $K(x, y) + F(x - y) + \\int_x^\\infty K(x, z) F(z - y) \\, dz = 0$ has kernels $F(x - y)$ and $F(z - y)$ that depend on differences; in the Gelfand–Levitan–Marchenko equation they must depend on the sums $x + y$ and $z + y$.\n\nSeveral conventions are in use, with other signs and with data on the left instead of the right.\nThe Gelfand–Levitan equation of [[Gelfand–Levitan theory|R4MRNP9EQT]] is a related equation, with an integral over $(0, x)$, that recovers a potential on a half-line from its spectral function.\n",
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