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  "body": "\n## Statement\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]].\nLet $A$ and $B$ be [[self-adjoint operators|0NEGN5CRSC]] on $H$, with domains $D(A)$ and $D(B)$.\nAssume that $A + B$ is [[essentially self-adjoint|DT49GKR9AS]] on $D(A) \\cap D(B)$, and let $C$ be its closure.\nThen for every $t \\in \\R$ and every $x \\in H$,\n$$e^{itC}x = \\lim_{n \\to \\infty} \\left( e^{itA/n} e^{itB/n} \\right)^n x,$$\nwhere the unitary groups $e^{itA}$, $e^{itB}$ and $e^{itC}$ are those of [[Stone's theorem|DVGQB92HPJ]].\n\n## Remarks\n\nFor $n \\times n$ matrices this is the [[Lie product formula|HK3BRMK912]], and the proof idea is the same.\nThe difference $e^{isA}e^{isB} - e^{isC}$ is $o(s)$ as $s \\to 0$ when applied to a vector of $D(A) \\cap D(B)$.\nA telescoping sum turns $n$ such errors of size $o(1/n)$ into a total error that tends to $0$.\nIn infinite dimensions, the convergence is in the strong sense, vector by vector, and the essential self-adjointness of $A + B$ is used to pass from $D(A) \\cap D(B)$ to general $x$.\nA similar formula holds for the semigroups $e^{-tA}$ and $e^{-tB}$ with $t \\ge 0$, when $A$ and $B$ are bounded below and $A + B$ is essentially self-adjoint on $D(A) \\cap D(B)$.\n\nExample: in quantum mechanics with $A = -\\Delta$ and $B = V$ a suitable potential, $e^{-itA/n}$ acts by a Fourier multiplier and $e^{-itV/n}$ by multiplication.\nThe formula approximates the Schrödinger evolution by alternating free motion and potential kicks; this is the basis of split-step methods.\n\n## Explanations\n\n### L1 human\n\nFor numbers, $e^{a + b} = e^{a}e^{b}$, but for operators that do not commute this fails.\nThe product formula repairs this: alternating many small steps of $A$ and of $B$ gives the evolution of $A + B$ in the limit.\nIt is like crossing a field diagonally by many tiny steps east and north in turn.\n\n### L2 human\n\nThe result is due to Trotter, who extended Lie's formula for matrices to unbounded operators.\nIt is used to build the Feynman–Kac formula: for $-\\Delta + V$, the alternating products become integrals over paths of Brownian motion.\nIn numerical analysis it justifies operator splitting, where a hard evolution is approximated by easy ones.\nThe hypothesis that $A + B$ is essentially self-adjoint on $D(A) \\cap D(B)$ ensures that the limit is the evolution of a well-defined Hamiltonian.\n",
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  "msc": [
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  "name": "Lie–Trotter product formula",
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  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/KH7/KH7EPKDCH3.md",
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  "url": "https://math2.tamnd.com/wiki/lie-trotter-product-formula/",
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