Bargmann transform

Summary

The Bargmann transform is a unitary map from $L^2(\R)$ onto the Segal–Bargmann space, the Hilbert space of entire functions $F$ on $\C$ with $\int_\C \abs{F(z)}^2 e^{-\abs{z}^2}\, dA(z) < \infty$. It sends the Hermite functions to the normalized monomials $z^n / \sqrt{n!}$. It turns the creation and annihilation operators of the harmonic oscillator into multiplication by $z$ and differentiation in $z$. This makes the oscillator very simple. It is closely related to the short-time Fourier transform with a Gaussian window.