Arithmetical chaos and violation of universality in energy level statistics

Abstract

A class of strongly chaotic systems revealing a strange arithmetical structure is discussed whose quantal energy levels exhibit level attraction rather than repulsion. As an example, the nearest-neighbor level spacings for Artin's billiard have been computed in a large energy range. It is shown that the observed violation of universality has its root in the existence of an infinite number of Hermitian operators (Hecke operators) which commute with the Hamiltonian and generate nongeneric correlations in the eigenfunctions.PACS numbers: 05.45.+b, 03.65.-w There seems to be no doubt that the statistical properties of quantal energy spectra and eigenvectors of classically chaotic systems are well described (with respect to their short-range correlations) by the universal laws of random-matrix theory (RMT) [1], originally proposed by Wigner and by Landau and Smorodinsky and fully developed by Dyson for a better understanding of the resonances of compound nuclei. (See Ref.[2] for a collection of the original papers, and Refs. [1,3,4] for recent reviews.) In the simplest case of generic Hamiltonian systems whose classical motion is time-reversal invariant, two universality classes are predicted: All systems with an integrable classical limit fall into one class, and all systems whose classical limit is strongly chaotic iK systems) fall into another class. Consider, e.g., the density distribution Pis) of "unfolded" nearest-neighbor level spacings s. In accordance with the empirical observation that the short-range spectral fluctuations of integrable systems are just those of random numbers, the level spacing distributions for these systems are Poissonian, i.e., ^PoissonOO = e ~\ while those of chaotic systems are the same as for the eigenvalues of large real symmetric random matrices, i.e., of the Gaussian orthogonal ensemble (GOE), which in the case of Pis) is well approximated by Wigner's surmise Pwigner(^) -J nse ~K S /4 . For small spacings, $-*0, the two distributions exhibit the wellknown phenomenon of level attraction [P(s) -\ -s] for classically integrable systems and of level repulsion [Pis)~~ j s] for strongly chaotic systems. It thus appears that the statistical properties of a given quantum system are already determined by its classical limit, depending only upon whether this is chaotic or not.In this Letter we study an interesting class of strongly chaotic systems whose quantal eigenvalues exhibit surprisingly enough level attraction rather than repulsion, and which thus lead to an apparent violation of the universal laws of RMT. We show that these systems reveal a strange arithmetical structure of chaos, which we call arithmetical chaos, that manifests itself in the existence of an infinite number of Hermitian operators commuting with the quantum Hamiltonian, and which are the origin of unexpected correlations in the quantum eigenstates. The existence of such operators could not be anticipated since these systems, having 2 degrees of freedom and being ergodic, possess classically only a single constant of m...

Numerical study of length spectra and low-lying eigenvalue spectra of compact hyperbolic 3-manifolds

“…see [2] and other articles therein). Because any classical dynamical systems of a free particle in CH spaces are strongly chaotic (K-systems), the semiclassical behavior of the statistical property of eigenvalues and eigenfunctions in these spaces has been intensively studied [3][4][5][6][7][8][9][10]. It is the Gutzwiller trace formula [11] that relates a set of periodic orbits(=geodesics) to a set of energy eigenstates and gives the semiclassical correspondence for classically chaotic systems.…”

“…Eigenvalues of CH 2-spaces has been numerically obtained by many authors [3][4][5]7,8,13]. Eigenvalues of cusped arithmetic and cusped non-arithmetic 3-manifolds with finite volume have been obtained by Grunewald and Huntebrinker using a finite element method [14].…”

Quantum and Arithmetical Chaos

“…The point is that integrable systems are systems with sufficiently large number of independent commuting operators and Hecke operators may be viewed as a manifestation of a kind of arithmetic integrability of arithmetic systems which does the Poisson statistics for these models natural [27]. Unfortunately, precise relations along this line seem to be impossible.…”

“…Consider these equations as transformations from variables t i to new variables m i [27]. The volume elements in these two representations are related as…”

Maximum norms of chaotic quantum eigenstates and random waves

“…An upper bound for Hecke eigenfunctions on certain three-dimensional arithmetic manifolds has been derived by Koyama and is given by ψ n ∞ < c ε E 37/76+ε n , ∀ ε > 0 [25]. In the case of two-dimensional systems with arithmetic chaos [16,[26][27][28][29][30] the upper bound Eq. (6) could be improved for a Hecke basis by Iwaniec and Sarnak [23] …”