Resonant eigenstates for a quantized chaotic system

Abstract

We study the spectrum of quantized open maps as a model for the resonance spectrum of quantum scattering systems. We are particularly interested in open maps admitting a fractal repeller. Using the ‘open baker's map’ as an example, we numerically investigate the exponent appearing in the fractal Weyl law for the density of resonances; we show that this exponent is not related with the ‘information dimension’, but rather the Hausdorff dimension of the repeller. We then consider the semiclassical measures associated with the eigenstates: we prove that these measures are conditionally invariant with respect to the classical dynamics. We then address the problem of classifying semiclassical measures among conditionally invariant ones. For a solvable model, the ‘Walsh-quantized’ open baker's map, we manage to exhibit a family of semiclassical measures with simple self-similar properties.