Distribution of matrix elements of chaotic systems

TLDR: In this article, it was shown that quantum perturbation theory must fail, for chaotic systems, in the semiclassical limit ε ≥ 0, for two arbitrarily close Hamiltonians with different sets of eigenvectors.

Abstract: When a quantum system has a chaotic classical analog, its matrix elements in the energy representation are closely related to various microcanonical averages of the classical system. The diagonal matrix elements cluster around the classical expectation values, with fluctuations similar to the values of the off-diagonal matrix elements. The latter in turn are related to the classical autocorrelations. These results imply that quantum perturbation theory must fail, for chaotic systems, in the semiclassical limit \ensuremath{\Elzxh}\ensuremath{\rightarrow}0: Two arbitrarily close Hamiltonians have, in general, completely different sets of eigenvectors.