Abstract
Consider the non-compact billiard in the first quandrant bounded by the positive x-semiaxis, the positive y-semiaxis and the graph of f(x)=(x+1)−α, α∈(1,2]. Although the Schnirelman Theorem holds, the quantum average of the position xis finite on any eigenstate, while classical ergodicity entails that the classical time average of xis unbounded.
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Graffi, S., Lenci, M. Localization in Infinite Billiards: A Comparison Between Quantum and Classical Ergodicity. Journal of Statistical Physics 116, 821–830 (2004). https://doi.org/10.1023/B:JOSS.0000037218.05161.f3
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DOI: https://doi.org/10.1023/B:JOSS.0000037218.05161.f3