The Correlation Between Multiplicities¶of Closed Geodesics on the Modular Surface

Peter, Manfred

doi:10.1007/s002201000574

Cited by 8 publications

(21 citation statements)

References 19 publications

(28 reference statements)

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“…We are now able to state our main theorems: which is the mean square of the weighted multiplicities for the modular domain (Q = 1), proved by M. Peter [10], following a conjecture of Bogomolny et al [3]. .…”

Section: Introductionmentioning

confidence: 94%

“…And we can see that for c > 2, A Q (q c ) does not depend on parity of c. Now we have by [10] In the second step we change L-function in the last formula by the Euler's product formula, and define…”

Section: Calculating the Mean Square Of Weighted Multiplicities Functionmentioning

confidence: 99%

“…We have the local factor decomposition for this function [10], [9]: For the next result we use the uniform limit periodicity of β (p,R) and once again (see lemma 6.10) get the formula…”

Section: Calculating the Mean Square Of Weighted Multiplicities Functionmentioning

confidence: 99%

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A mean value theorem for closed geodesics on congruence surfaces

Lukianov

2007

Forum Mathematicum

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Abstract. We define a weighted multiplicity function for closed geodesics of given length on a finite area Riemann surface. These weighted multiplicities appear naturally in the Selberg trace formula, and in particular their mean square plays an important role in the study of statistics of the eigenvalues of the Laplacian on the surface.In the case of the modular domain, E. Bogomolny, F. Leyvraz and C. Schmit gave a formula for the mean square, which was rigorously proved by M. Peter. In this paper we calculate the mean square of weighted multiplicities for some surfaces associated to congruence subgroups of the unit group of a rational quaternion algebra, in particular for congruence subgroups of the modular group. Remarkably, the result turns out to be a rational multiple of the mean square for the modular domain.

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Section: Introductionmentioning

confidence: 94%

Section: Calculating the Mean Square Of Weighted Multiplicities Functionmentioning

confidence: 99%

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A mean value theorem for closed geodesics on congruence surfaces

Lukianov

2007

Forum Mathematicum

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“…The above approach implies that the 2-point function converges to that of Poisson-distributed points, quite different from RMT [6]; this untypical statistics has been checked numerically, but a full proof of is still lacking (mainly due to the "second problem" mentioned above) [25].…”

Section: Spectral Statisticsmentioning

confidence: 99%

Some open questions in ‘wave chaos’

Nonnenmacher¹

2008

Nonlinearity

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The subject area referred to as "wave chaos", "quantum chaos" or "quantum chaology" has been investigated mostly by the theoretical physics community in the last 30 years. The questions it raises have more recently also attracted the attention of mathematicians and mathematical physicists, due to connections with number theory, graph theory, Riemannian, hyperbolic or complex geometry, classical dynamical systems, probability etc. After giving a rough account on "what is quantum chaos?", I intend to list some pending questions, some of them having been raised a long time ago, some others more recent. The choice of problems (and of references) is of course partial and personal. 1 0.1. A brief overview of "quantum chaos". Let us start by resolving the apparent paradox of the phrase "quantum chaos". A classically chaotic system is defined by an ordinary differential equation, or a discrete-time map acting on some finite-dimensional phase space; for the dynamics to be chaotic, the ODE or the map is generally nonlinear in the phase space coordinates. By contrast, the associated quantum system is defined by a linear operator acting on the quantum Hilbert space; the quantum dynamics generated by this operator (either a selfadjoint Hamiltonian, or a unitary propagator) is by essence linear, but acts in a space of infinite (or at least large) dimension. Let us assume that the spectrum of that operator is discrete: the quantum dynamics is then quasiperiodic, and can be easily expressed in terms of the eigenvalues and eigenstates of the operator. One is thus led, from a time-dependent (dynamical) problem, to a time-independent (spectral) problem, which forms the "backbone" of quantum chaos:What are the spectral properties of a quantum Hamiltonian, the classical limit of which generates a chaotic dynamics?What do the corresponding eigenstates look like?These two questions are deliberately vague. Below I will give more precise formulations. To fix ideas, we will consider one of the simplest chaotic systems, namely the geodesic flow on a compact Riemannian surface (X, g) of negative curvature (and unit area). The corresponding quantum Hamiltonian is defined as the generator of the wave group on X, 1 This contribution was mostly written while I enjoyed a stay at the Centre de Recherches Mathématiques in Montréal, invited by D.Jakobson and I.Polterovich, to whom I am thankful.

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“…The main result in this paper is to propose asymptotic formulas for the sum of higher shifted product of the multiplicities in length spectrum for any congruence subgroup of the modular group with an explicit description of the coefficient of their leading term, which is an extension of the results in [3,23,17]. Since the multiplicities are given by the class numbers of quadratic forms, they are approximated by periodic functions [23].…”

Section: Introductionmentioning

confidence: 99%

Correlations of multiplicities in length spectra for congruence subgroups

Hashimoto

2012

Bulletin of the London Mathematical Society

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Bogomolny- Leyvraz-Schmit (1996) and Peter (2002) proposed an asymptotic formula for the correlation of the multiplicities in length spectrum on the modular surface, and Lukianov (2007) extended its asymptotic formula to the Riemann surfaces derived from the congruence subgroup Γ 0 (n) and the quaternion type co-compact arithmetic groups. The coefficients of the leading terms in these asymptotic formulas are described in terms of Euler products over prime numbers, and they appear in eigenvalue statistic formulas found by Rudnick (2005) and Lukianov (2007) for the Laplace-Beltrami operators on the corresponding Riemann surfaces. In the present paper, we further extend their asymptotic formulas to the higher level correlations of the multiplicities for any congruence subgroup of the modular group.

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The Correlation Between Multiplicities¶of Closed Geodesics on the Modular Surface

Cited by 8 publications

References 19 publications

A mean value theorem for closed geodesics on congruence surfaces

A mean value theorem for closed geodesics on congruence surfaces

Some open questions in ‘wave chaos’

Correlations of multiplicities in length spectra for congruence subgroups

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