The sharp Heisenberg theorem is owned by functional analysis

Assume countable choice, as does the cited theorem. The sharp Heisenberg uncertainty inequality and its equality classification are owned by functional analysis: the published theorem Heisenberg uncertainty and Gaussian equality of the functional-analysis track states that for $f\in \mathcal{S}({\mathbb{R}}^n)$ and $a,b\in {\mathbb{R}}^n$ $\parallel \mathrm{\mid }x-a\mathrm{\mid }f{\parallel }_2\parallel \mathrm{\mid }\xi -b\mathrm{\mid }\widehat{f}{\parallel }_2\ge \frac{n}{4\pi }\mathrm{\parallel }f{\mathrm{\parallel }}_2^2,$ with equality for nonzero $f$ exactly for $f(x)=c\exp (-\lambda \mathrm{\mid }x-a{\mathrm{\mid }}^2\mathrm{/}2)\exp (2\pi ib\cdot x)$, $c\mathrm{\ne }0$, $\lambda >0$. Both this inequality and its sharp equality classification are quoted here only on the Schwartz domain of the published source. Under this library's $e^{-2\pi ix\cdot \xi }$ convention its coordinate form on that domain is $\mathrm{\parallel }{x}_{j}f{\mathrm{\parallel }}_2\mathrm{\parallel }{\xi }_j\widehat{f}{\mathrm{\parallel }}_2\ge (4\pi {)}^{-1}\mathrm{\parallel }f{\mathrm{\parallel }}_2^2$.

This page does not extend the source's equality classification beyond Schwartz functions. Separately, the local cutoff argument in The coordinate inequality $\mathrm{\parallel }{x}_{j}f{\mathrm{\parallel }}_2\mathrm{\parallel }{D}_{j}f{\mathrm{\parallel }}_2\ge \frac{1}{2}\mathrm{\parallel }f{\mathrm{\parallel }}_2^2$ supplies the coordinate real-variable inequality on the natural $H^1$ domain with $xf\in L^2$, Centring by translation and modulation preserves the variance product centres the variance formulation, and The n-dimensional Heisenberg uncertainty inequality records the summed $n$-dimensional inequality on that same domain. Gaussian attainment in this convention is checked in The Gaussian attains equality in the Heisenberg inequality ↗.