Spectral properties and dynamics of quantized Henon maps

We study a generalization of the Airy function, and use its properties to investigate the dynamics and spectral properties of the unitary operators on $L^2(\mathbf{R})$ of the form $U_c:=Fe^{i(q(x)+cx)}$, where $q$ is a real polynomial of odd degree, $c$ is a real number, and $F$ is the Fourier transform. We show that $U_c$ is a quantization of the classical Henon map

and show that for $c>0$ sufficiently large, $U_c$ has purely continuous spectrum. This fact has implications for the dynamics of $U_c$, which are shown to correspond when the condition is satisfied to the dynamics of its classical counterpart on $\mathbf{R}^2$.