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  "body": "\n## Statement\n\nLet $n \\ge 1$.\nFor $f \\in L^2(\\R^n)$ and $t \\in \\R$, let $e^{it\\Delta} f$ be the solution at time $t$ of the free [[Schrödinger equation|EPZR6H2WJN]] $i \\partial_t u + \\Delta u = 0$ with $u(0) = f$.\nIt is defined by the [[Fourier transform|DPVM34ADZV]] $\\hat f(\\xi) = \\int_{\\R^n} f(x) e^{-i x \\cdot \\xi} \\, dx$ as $\\widehat{e^{it\\Delta} f}(\\xi) = e^{-it\\abs{\\xi}^2} \\hat f(\\xi)$.\nThen for all $f \\in L^1(\\R^n) \\cap L^2(\\R^n)$ and all $t \\ne 0$,\n$$\\norm{e^{it\\Delta} f}_{L^\\infty(\\R^n)} \\le (4\\pi \\abs{t})^{-n/2} \\norm{f}_{L^1(\\R^n)}.$$\n\n## Remarks\n\nFor $t \\ne 0$ the solution operator has an explicit kernel:\n$$e^{it\\Delta} f(x) = (4\\pi i t)^{-n/2} \\int_{\\R^n} e^{i \\abs{x - y}^2 / (4t)} f(y) \\, dy,$$\nwith the principal branch of the square root.\nThe kernel has modulus $(4\\pi\\abs{t})^{-n/2}$ at every point, and the estimate follows at once.\nThe formula for the kernel comes from the Fourier transform of the complex Gaussian $e^{-it\\abs{\\xi}^2}$, or from the heat kernel by replacing $t$ by $it$.\n\nThe solution also conserves the $L^2$ norm: $\\norm{e^{it\\Delta} f}_{L^2} = \\norm{f}_{L^2}$, by Plancherel's theorem.\nInterpolation between this and the dispersive estimate with the Riesz–Thorin theorem gives, for $2 \\le r \\le \\infty$ and $1/r + 1/r' = 1$,\n$$\\norm{e^{it\\Delta} f}_{L^r} \\le (4\\pi\\abs{t})^{-n(1/2 - 1/r)} \\norm{f}_{L^{r'}}.$$\n\nThe decay rate $\\abs{t}^{-n/2}$ cannot be improved.\nFor the Gaussian $f(x) = e^{-\\abs{x}^2}$ one computes $\\abs{e^{it\\Delta} f(0)} = (1 + 16 t^2)^{-n/4}$, which is of exact order $\\abs{t}^{-n/2}$ for large $\\abs{t}$.\nThe $L^1$ norm on the right cannot be replaced by the $L^2$ norm.\nFor fixed $t$, the operator $e^{it\\Delta}$ maps $L^2(\\R^n)$ onto $L^2(\\R^n)$, which contains unbounded functions, so no bound of $\\norm{e^{it\\Delta} f}_{L^\\infty}$ by a multiple of $\\norm{f}_{L^2}$ holds.\n\n## Explanations\n\n### L1 human\n\nA free quantum particle described by the Schrödinger equation spreads out over time.\nWaves of different frequencies travel at different speeds, so an initially concentrated wave packet disperses.\nThe estimate says how fast: in $n$ dimensions, the peak height of the wave decays like $\\abs{t}^{-n/2}$, as long as the initial wave has finite total mass in $L^1$.\n\n### L2 human\n\nThe dispersive estimate is the pointwise decay input for the [[Strichartz estimates|4PDKJ8N1S9]].\nCombined with the $TT^*$ argument and the Hardy–Littlewood–Sobolev inequality, it gives space-time integrability of $e^{it\\Delta} f$ for $L^2$ data.\nThese bounds are the basic tool of the local and global well-posedness theory of nonlinear Schrödinger equations.\nThe same pattern applies to other [[dispersive equations|AYMJ2NFR4W]] such as the wave and Airy equations, with different decay rates.\nOne proves a pointwise decay bound from an oscillatory kernel and interpolates it with energy conservation.\n",
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