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  "body": "\n## Statement\n\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]], and let $A$ be a [[densely defined operator|BCJHNHKQFG]] on $H$ with domain $D(A)$.\nAssume that $A$ is self-adjoint: $A = A^*$, with equal domains, where $A^*$ is the [[adjoint|86B7HF79PQ]] of $A$.\nThen there is a unique [[projection-valued measure|ATGT25E5FF]] $E$ on the [[Borel sets|0WQXY32Q8D]] of $\\R$ such that, with $E_{x,y}(B) = \\ip{E(B) x}{y}$:\n\n1. $D(A) = \\set{x \\in H : \\int_{\\R} \\lambda^2 \\, d E_{x,x}(\\lambda) \u003c \\infty}$;\n2. $\\ip{A x}{y} = \\int_{\\R} \\lambda \\, d E_{x,y}(\\lambda)$ for all $x \\in D(A)$ and $y \\in H$.\n\nMoreover, $E(\\R \\setminus \\sigma(A)) = 0$, where $\\sigma(A) \\subseteq \\R$ is the [[spectrum|BMKATJ802A]] of $A$.\n\n## Remarks\n\nOne proof uses the Cayley transform $U = (A - iI)(A + iI)^{-1}$, a unitary operator for which $1$ is not an eigenvalue.\nThe [[spectral theorem for bounded normal operators|X71XRPJR7D]] applied to $U$ gives a projection-valued measure on the unit circle.\nIt is carried over to $\\R$ by the inverse of the map $\\lambda \\mapsto (\\lambda - i)/(\\lambda + i)$.\nEquivalently, if $H$ is separable, $A$ is unitarily equivalent to the multiplication by a real measurable function $\\varphi$ on some $L^2(\\Omega, \\mu)$, on the domain of all $f$ with $\\varphi f \\in L^2(\\Omega, \\mu)$.\nExample: on $L^2(\\R)$, the momentum operator $-i \\, d/dx$ on the Sobolev space $H^1(\\R)$ is self-adjoint; the Fourier transform turns it into multiplication by the variable $\\xi$, with spectrum $\\R$.\nSelf-adjointness, not mere symmetry, is needed.\nThe operator $-i \\, d/dx$ on smooth functions with compact support in $(0, \\infty)$ is symmetric, but it has no self-adjoint extension and no spectral resolution of this form.\n\n## Explanations\n\n### L1 human\n\nMany important operators, such as derivatives, are unbounded: they can make vectors arbitrarily long.\nThe theorem says that a self-adjoint unbounded operator is still, in a precise sense, a real diagonal operator.\nIts eigenvalues are replaced by a measure on the real line whose values are projections.\n\n### L2 human\n\nThe theorem is the mathematical basis of quantum mechanics, where observables such as position, momentum and energy are unbounded self-adjoint operators.\nIt defines $f(A) = \\int f(\\lambda) \\, dE(\\lambda)$ for Borel functions $f$, in particular the unitary group $e^{itA}$ that gives the time evolution of a quantum system.\nThe probability that a measurement of $A$ in the unit state $x$ gives a value in $B$ is $\\ip{E(B) x}{x}$.\nStone's theorem gives the converse correspondence between strongly continuous one-parameter unitary groups and self-adjoint operators.\n",
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