Bargmann transform

The Bargmann transform of a function, $f$, is a linear map $B:X(ℝ)\to Y(ℂ)$ defined by

$$Bf(z)=\sqrt{2}{\int }_{ℝ}f(t)e^{2\pi tz-\pi t^2-\frac{\pi }{2}{z}^2}𝑑t$$

Theorem.

The Bargmann transform on $L^{\mathrm{2}}\mathit{}\mathrm{(}\mathrm{R}\mathrm{)}$, $B\mathrm{:}{L}^{\mathrm{2}}\mathit{}\mathrm{(}\mathrm{R}\mathrm{)}\mathrm{\to }{\mathrm{F}}^{\mathrm{2}}\mathit{}\mathrm{(}\mathrm{C}\mathrm{)}$, is a unitary transformation. Here ${\mathrm{F}}^{\mathrm{2}}\mathit{}\mathrm{(}\mathrm{C}\mathrm{)}$ is the Fock space.

Title	Bargmann transform
Canonical name	BargmannTransform
Date of creation	2013-03-22 16:44:45
Last modified on	2013-03-22 16:44:45
Owner	ErlendA (6587)
Last modified by	ErlendA (6587)
Numerical id	7
Author	ErlendA (6587)
Entry type	Definition
Classification	msc 43A15
Defines	Bargmann transform