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Localization in infinite billiards: a comparison between quantum and classical ergodicity

Sandro Graffi·Marco Lenci
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Abstract

Consider the non-compact billiard in the first quandrant bounded by the positive $x$-semiaxis, the positive $y$-semiaxis and the graph of $f(x) = (x+1)^{-α}$, $α\in (1,2]$. Although the Schnirelman Theorem holds, the quantum average of the position $x$ is finite on any eigenstate, while classical ergodicity entails that the classical time average of $x$ is unbounded.

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Sandro Graffi, Marco Lenci. 2003-06-28. Localization in infinite billiards: a comparison between quantum and classical ergodicity. https://doi.org/10.1023/b%3Ajoss.0000037218.05161.f3

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