Tobias Weich

Experimental observation of the spectral gap in microwave n-disk systems.

Symmetry reduced three-disk and five-disk systems are studied in a microwave setup. Using harmonic inversion the distribution of the imaginary parts of the resonances is determined. With increasing opening of the systems, a spectral gap is observed for thick as well as for thin repellers and for the latter case it is compared with the known topological pressure bounds. The maxima of the distributions are found to coincide for a large range of the distance to radius parameter with half of the classical escape rate. This confirms theoretical predictions based on rigorous mathematical analysis for the spectral gap and on numerical experiments for the maxima of the distributions.

Symmetry reduction of holomorphic iterated function schemes and factorization of Selberg zeta functions

Given a holomorphic iterated function scheme with a finite symmetry group

Classical and quantum resonances for hyperbolic surfaces

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Weyl asymptotics: from closed to open systems.

, we show that the associated dynamical zeta function factorizes into symmetry-reduced analytic zeta functions that are parametrized by the unitary irreducible representations of

Asymptotic spectral gap and Weyl law for Ruelle resonances of open partially expanding maps

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Resonance Chains and Geometric Limits on Schottky Surfaces

. We show that this factorization implies a factorization of the Selberg zeta function on symmetric

Resonance chains in open systems, generalized zeta functions and clustering of the length spectrum

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Formation and interaction of resonance chains in the open three-disk system

-funneled surfaces and that the symmetry factorization simplifies the numerical calculations of the resonances by several orders of magnitude. As an application this allows us to provide a detailed study of the spectral gap and we observe for the first time the existence of a macroscopic spectral gap on Schottky surfaces.

On the support of Pollicott-Ruelle resonanant states for Anosov flows

For compact and for convex co-compact oriented hyperbolic surfaces, we prove an explicit correspondence between classical Ruelle resonant states and quantum resonant states, except at negative integers where the correspondence involves holomorphic sections of line bundles.

Improved fractal Weyl bounds for hyperbolic manifolds

We present microwave experiments on the symmetry reduced five-disk billiard studying the transition from a closed to an open system. The measured microwave reflection signal is analyzed by means of the harmonic inversion and the counting function of the resulting resonances is studied. For the closed system this counting function shows the Weyl asymptotic with a leading exponent equal to 2. By opening the system successively this exponent decreases smoothly to a noninteger value. For the open systems the extraction of resonances by the harmonic inversion becomes more challenging and the arising difficulties are discussed. The results can be interpreted as a first experimental indication for the fractal Weyl conjecture for resonances.