×

On the infinitesimal generator of the Fourier transform. (English) Zbl 0759.42007

In the group \(U(L^ 2(\mathbb{R}))\) for all unitary operators in \(L^ 2(\mathbb{R})\) the Fourier transform can be joined with the identity by means of a 1-parameter subgroup. The author uses the irreducibility of the Schrödinger representation to prove that \(H:=i{\pi\over 4}\left[{d^ 2\over dx^ 2}-x^ 2+id\right]\) is the infinitesimal generator of this subgroup.

MSC:

42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
47D03 Groups and semigroups of linear operators
PDFBibTeX XMLCite