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Test of Trace Formulas for Spectra of Superconducting Microwave Billiards

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Abstract

Experimental tests of various trace formulas, which in general relate the density of states for a given quantum mechanical system to the properties of the periodic orbits of its classical counterpart, for spectra of superconducting microwave billiards of varying chaoticity are reviewed by way of examples. For a two-dimensional Bunimovich stadium billiard the application of Gutzwiller's trace formula is shown to yield correctly locations and strengths of the peaks in the Fourier transformed quantum spectrum in terms of the shortest unstable classical periodic orbits. Furthermore, in two-dimensional billiards of the Limaçon family the transition from regular to chaotic dynamics is studied in terms of a recently derived general trace formula by Ullmo, Grinberg and Tomsovic. Finally, some salient features of wave dynamical chaos in a fully chaotic three-dimensional Sinai microwave billiard are discussed. Here the reconstruction of the spectrum is not as straightforward as in the two-dimensional cases and a modified trace formula as suggested by Balian and Duplantier will have eventually to be applied.

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  1. Institut für Kernphysik, Technische Universität Darmstadt, D-64289, Darmstadt, Germany

    A. Richter

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Richter, A. Test of Trace Formulas for Spectra of Superconducting Microwave Billiards. Foundations of Physics 31, 327–354 (2001). https://doi.org/10.1023/A:1017594604474

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