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  "body": "\n## Definition\n\nLet $I \\subseteq \\R$ be an open interval and let $V \\colon I \\to \\R$ be locally integrable.\nThe one-dimensional Schrödinger operator with potential $V$ is the differential expression\n$$H y = -y'' + V y,$$\nacting on functions $y \\colon I \\to \\C$ such that $y$ and $y'$ are locally absolutely continuous.\nMore generally, for an open set $\\Omega \\subseteq \\R^d$ and a real function $V$ on $\\Omega$, the Schrödinger operator is $H = -\\Delta + V$, where $\\Delta$ is the [[Laplacian|0E0B6RA8K0]].\nThe equation $H y = \\lambda y$ with $\\lambda \\in \\C$ is the time-independent Schrödinger equation.\n\n## Remarks\n\nExample: for $I = \\R$ and $V(x) = x^2$, the function $y(x) = e^{-x^2/2}$ has $y'' = (x^2 - 1) e^{-x^2/2}$.\nSo $H y = -y'' + x^2 y = y$, and $y$ solves the Schrödinger equation with $\\lambda = 1$.\nNon-example: $-y'' + i x y$ is not a Schrödinger operator in this sense, because the potential $i x$ is not real.\nIn units where Planck's constant and the mass are normalised, $H$ is the energy operator of a quantum particle.\nTo get a self-adjoint operator on $L^2(I)$ one must also fix boundary conditions at the ends of $I$ when they are needed.\n\n## Explanations\n\n### L1 human\n\nA Schrödinger operator is minus the second derivative plus multiplication by a real function.\nIt is the energy of a quantum particle that moves in a landscape of height $V$.\n",
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