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  "body": "\n## Summary\n\nA Schrödinger operator is a differential operator of the form $-\\Delta + V$, where $\\Delta$ is the [[Laplacian|0E0B6RA8K0]] and $V$ is a real function called the potential.\nIn one dimension it is the operator $-\\frac{d^2}{dx^2} + V(x)$.\nIts eigenvalue equation is the time-independent Schrödinger equation of quantum mechanics, and its eigenvalues are the allowed energies of a particle in the potential $V$.\nIn the theory of ordinary differential equations it is the main example of a Sturm–Liouville operator.\n\n## Explanations\n\n### L1 human\n\nThe operator takes a function, measures how much it bends with the second derivative, and adds the function times the potential.\nIn physics the bending term is the kinetic energy and $V$ is the potential energy.\nFor $V(x) = x^2$ the operator $-y'' + x^2 y$ is the quantum harmonic oscillator.\nIts eigenvalues are $1, 3, 5, \\ldots$, and the first eigenfunction is $e^{-x^2/2}$.\n",
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