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We resolve a long-standing riddle in quantum chaos, posed by certain fullychaotic billiards with constant negative curvature whose periodic orbits arehighly degenerate in length. Depending on the boundary conditions for thequantum wave functions, the energy spectra either have uncorrelated levelsusually associated with classical integrability or conform to the "universal"Wigner-Dyson type although the classical dynamics in both cases is the same.The resolution turns out surprisingly simple. The Maslov indices of orbitswithin multiplets of degenerate length either yield equal phases for therespective Feynman amplitudes (and thus Poissonian level statistics) or giverise to amplitudes with uncorrelated phases (leading to Wigner-Dyson levelcorrelations). The recent semiclassical explanation of spectral universality inquantum chaos is thus extended to the latter case of "pseudo-arithmetical"chaos.

Level statistics in arithmetical and pseudo-arithmetical chaos