Spectral properties and dynamics of quantized Henon maps

Abstract:

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 \begin{align*} f_\lambda :\mathbf {R}^2 &\to \mathbf {R}^2 , (x,y) &\mapsto (y+q’(x)+c,-x), \end{align*} 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 $\mathbf {R}^2$.