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Resonant eigenstates in quantum chaotic scattering

Stéphane Nonnenmacher·Mathieu Rubin
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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.

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Stéphane Nonnenmacher, Mathieu Rubin. 2007-04-23. Resonant eigenstates in quantum chaotic scattering. https://arxiv.org/abs/nlin/0608069

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