Abstract. We investigate the dynamics and spectral properties of the unitary operators $U_\lambda :=e^{i\lambda x^2}F$, where $\lambda\in {\msbm R}$ and $F$ is the Fourier transform. We show that $U_\lambda $ is a quantization of the classical map $$ f_\lambda\colon {\msbm R}^2 \to{\msbm R}^2 (x,y) \mapstochar\rightarrow (y,2\lambda y-x), $$ and that the phase transition at $|\lambda |=1$ for $f_\lambda $ corresponds to a similar phase transition for $U_\lambda $, which changes at those values from having a pure point to a continuous spectrum.

AMS Subject Classification (1991): 32H50, 37N20

Keyword(s): Unitary dynamics

Received June 6, 2002, and in revised form November 13, 2002. (Registered under 2920/2009.)