Level statistics in arithmetical and pseudo-arithmetical chaos Petr Braun 1 , $^{2}$, Fritz Haake1 $^{1}$Fachbereich Physik, Universit¨ at Duisburg-Essen, 47048 Duisburg, Germany $^{2}$Department of Theoretical Physics, Saint-Petersburg University, 198504 Saint-Petersburg, Russia We resolve a long-standing riddle in quantum chaos, posed by certain fully chaotic billiards with constant negative curvature whose periodic orbits are highly degenerate in length. Depending on the boundary conditions for the quantum wave functions, the energy spectra either have uncorrelated levels usually 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 orbits within multiplets of degenerate length either yield equal phases for the respective Feynman amplitudes (and thus Poissonian level statistics) or give rise to amplitudes with uncorrelated phases (leading to Wigner-Dyson level correlations). The recent semiclassical explanation of spectral universality in quantum chaos is thus extended to the latter case of “pseudo-arithmetical” chaos. PACS numbers: 05.45.Mt, 03.65.Sq Introduction : After more than two decades of inves- tigations, the famous BGS conjecture [1] has recently found a semiclassical explanation[2–6]: The quantum level spectra of classically chaotic systems display fluctu- ations conforming to the random-matrix-theory (RMT) predictions for the Wigner-Dyson universality classes. However, there is the notable exception of “arithmeti- cal” systems which are fully chaotic classically but dis- play quantum spectral statistics close to Poissonian, a be- havior usually associated with integrable classical motion [7–12]. Quantum mechanically these exceptional dynam- ics exhibit an infinite number of the so called Hecke op- erators commuting with the Hamiltonian. Therefore the energy spectrum falls into practically independent mul- tiplets such that nearby levels bear no correlation. On the classical side the periodic-orbit action spectra of such systems are distinguished by a degeneracy exponentially growing with the orbit period. On the other hand, by merely changing the boundary conditions for the quantum wave functions of some such exceptional arithmetical systems one can retrieve univer- sal spectral fluctuations ` a la Wigner and Dyson while not at all changing the classical dynamics. It is customary to speak of pseudo-arithmetical systems then [13–15]. The strikingly different quantum behavior of arith- metical and pseudo-arithmetical systems might raise doubts about the validity of the recent semiclassical ex- planation of universal quantum spectral fluctuations un- der conditions of classical chaos; after all, the classical dynamics are identical for the systems under discussion, and so appear, on first sight, the Gutzwiller type semi- classical periodic-orbit expansions. Various suggestions were ventured for the effect of the boundary conditions, among them a distinction between the orbit classes con- tributing to the Selberg trace formula applicable (and exact) in the arithmetical case, and the Gutzwiller for- mula applicable in the pseudo-arithmetical case. We show here that the explanation is much simpler and lies in special properties of periodic orbits. Due to these peculiarities all equal-length orbits (save for a negligible fraction) of an arithmetical system contribute Feynman amplitudes with the same Maslov phase; their construc- tive interference makes for nonuniversal spectral statis- tics. In the pseudo-arithmetical case these phases vary randomly within a degenerate-action multiplet such that destructive interference makes the high action degener- acy ineffective. The difference between the two cases is most easily revealed for the diagonal approximation to the spectral form factor, and therefore that approxima- tion will play a central role here; off-diagonal corrections will be discussed briefly in the end. The billiard T ∗ (2 , 3 , 8): We shall not deal with the al- ternative arithmetical/pseudo-arithmetical in full gener- ality but prefer to work with a representative example, the so called triangular billiard T ∗ (2 , 3 , 8). That system was first considered in the studies of the free motion on the surface of constant negative curvature tesselated by regular octagons[16]. Desymmetrization of the regular octagon necessary to get rid of the rotational and re- flection symmetry leads to a triangular fundamental do- main with the angles π/ 2 , π/ 3 , π/ 8. Depicted inside the Poincar´ e disk | z | ≤ 1 in the complex plane, the triangle has its hypotenuse ( N ) and the longer leg ( L ) directed along two diameters whereas the shorter leg ( M ) looks like an arc (Fig. 1). A classical periodic orbit folded into the triangle looks like a sequence of arcs mirror- reflected from the sides of the triangle. The classical mo- tion is completely chaotic, all periodic orbits having the same Lyapunov constant. The multiplicities in the length spectrum of periodic orbits grow exponentially with the length, like exp( l$_{γ}$/ 2), where l$_{γ}$ is made dimensionless by referral to a scale fixed by setting the curvature to − 1; since action is proportional to the orbit length the action spectrum also has an exponentially growing degeneracy. The quantum energy levels for T ∗ (2 , 3 , 8) are found as eigenvalues of the Laplace-Beltrami operator. The arXiv:1001.3339v2 [nlin.CD] 1 Feb 2010