Crystal properties of eigenstates for quantum cat maps

Abstract

Using the Bargmann-Husimi representation of quantum mechanics on a torus phase space, we study analytically eigenstates of quantized cat maps. The linearity of these maps implies a close relationship between classically invariant sublattices on the one hand, and the patterns (or `constellations') of Husimi zeros of certain quantum eigenstates on the other hand. For these states, the zero patterns are crystals on the torus. As a consequence, we can compute explicit families of eigenstates for which the zero patterns become uniformly distributed on the torus phase space in the limit $\hslash \to 0$. This result constitutes a first rigorous example of semi-classical equidistribution for Husimi zeros of eigenstates in quantized one-dimensional chaotic systems.

43 pages, LaTeX, including 7 eps figures Some amendments were made in order to clarify the text, mainly in the 4 first sections. Figures are unchanged. To be published in: Nonlinearity

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