Resonant eigenstates in quantum chaotic scattering

Stéphane Nonnenmacher (1) , Mathieu Rubin (1)

1 SPhT - Service de Physique Théorique

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.

Mots clés: chaotic scattering; fractal Weyl law; resonant eigenstate; semiclassical measure

Domaines: Dynamique Chaotique [nlin.CD]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Stéphane Nonnenmacher, Mathieu Rubin. Resonant eigenstates in quantum chaotic scattering. Nonlinearity, 2007, 20 (6), pp.1387-1420. ⟨10.1088/0951-7715/20/6/004⟩. ⟨hal-00090428v3⟩