Fourier transforms and the dilatation group

Abstract

The Fourier transform g(k) of a square integrable function f(x), vanishing for x<0, is analytic in the upper half plane, so that, replacing k by k exp ζ, k≥0, 0<Im ζ<π, it can be associated with an operator K(ζ) in ℋ+=L2((0,∞),dx). The operator K(ζ) can be expressed in terms of the generator D of the dilatation group on ℋ+ and it can be shown that it is analytic in the strip 0<Im ζ<π with strong limits as Im ζ↓0 and ↑π. The Laplace transform (ζ=iπ/2) is an analytic vector for D. It is also found that D is not a spectral operator of scalar type on Lp((0,∞),dx), 1≤p<∞, p≠2. Applying the results obtained here to the time-evolution operator for a one-dimensional Sommerfeld model for the interaction between an electron and a metal, it is found that this operator has a complex-dilated analytic extension.