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  "body": "\n## Example\n\nConsider the stationary [[Schrödinger equation|EPZR6H2WJN]] $-\\Delta \\psi + V \\psi = k^2 \\psi$ on $\\R^3$, with a real potential $V$ of compact support, in units with $\\hbar^2 / 2m = 1$.\nLet $k \u003e 0$ and let $\\mathbf{k} \\in \\R^3$ with $\\abs{\\mathbf{k}} = k$ be the direction and wavenumber of an incoming plane wave.\nThe scattering solution $\\psi$ solves the Lippmann–Schwinger equation\n$$\\psi(x) = e^{i \\mathbf{k} \\cdot x} - \\frac{1}{4\\pi} \\int_{\\R^3} \\frac{e^{ik\\abs{x - y}}}{\\abs{x - y}} \\, V(y) \\, \\psi(y) \\, dy, \\quad x \\in \\R^3.$$\nIt is an [[integral equation of the second kind|8KJCN49V64]] for $\\psi$, with the [[weakly singular kernel|94TNBAAJYX]] $-\\frac{1}{4\\pi} \\frac{e^{ik\\abs{x - y}}}{\\abs{x - y}} V(y)$.\n\nCheck: the function $G(x) = -\\frac{e^{ik\\abs{x}}}{4\\pi \\abs{x}}$ is the outgoing [[Green's function|SH6JY9RN2E]] of the Helmholtz operator, $(\\Delta + k^2) G = \\delta_0$.\nSince $(\\Delta + k^2) e^{i\\mathbf{k} \\cdot x} = 0$, applying $\\Delta + k^2$ to the right side gives $V \\psi$, which is the Schrödinger equation $(\\Delta + k^2)\\psi = V \\psi$.\nThe choice of $e^{+ik\\abs{x - y}}$ makes the scattered part $\\psi - e^{i \\mathbf{k} \\cdot x}$ an outgoing spherical wave at infinity.\n\n## Remarks\n\nIn operator form the equation reads $\\psi = \\phi + G_0 V \\psi$ with $\\phi$ the incoming wave and $G_0$ the convolution with $G$.\nIterating it gives the [[Born series|2PCVRG9NBN]], and its first term beyond $\\phi$ is the Born approximation.\n\n## Explanations\n\n### L1 human\n\nA wave hits an obstacle described by $V$ and scatters.\nThe equation says that the total wave is the incoming wave plus the waves sent out by each point of the obstacle.\nEach point sends out a wave with strength equal to the potential times the total wave there.\nSince the unknown wave appears on both sides, it is an integral equation rather than a formula.\n",
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