Resonant eigenstates in quantum chaotic scattering

Details Resonant eigenstates in quantum chaotic scattering Resonant eigenstates in quantum chaotic scattering

2006-08-31

arxiv.org/abs/nlin/0608069v3

Nonlinear Sciences

Chaotic Dynamics

39 pages, 10 figures Compared with the previous version, we generalized the correspondence between semiclassical measures and

Scientific paper

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.

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