Jens Marklof

Arithmetic Quantum Chaos

The central objective in the study of quantum chaos is to characterize universal properties of quantum systems that reflect the regular or chaotic features of the underlying classical dynamics. Most developments of the past 25 years have been influenced by the pioneering models on statistical properties of eigenstates (Berry 1977) and energy levels (Berry and Tabor 1977; Bohigas, Giannoni and Schmit 1984). Arithmetic quantum chaos (AQC) refers to the investigation of quantum system with additional arithmetic structures that allow a significantly more extensive analysis than is generally possible. On the other hand, the special number-theoretic features also render these systems non-generic, and thus some of the expected universal phenomena fail to emerge. Important examples of such systems include the modular surface and linear automorphisms of tori (‘cat maps’) which will be described below. The geodesic motion of a point particle on a compact Riemannian surface M of constant negative curvature is the prime example of an Anosov flow, one of the strongest characterizations of dynamical chaos. The corresponding quantum eigenstates φj and energy levels λj are given by the solution of the eigenvalue problem for the LaplaceBeltrami operator ∆ (or Laplacian for short)

LIMIT THEOREMS FOR THETA SUMS

For almost all values of x 2 R, the classical theta sum S N (x) = N ?1=2 N X n=1 e 2i n 2 x exhibits an extremely irregular behaviour, as N tends to innnity. This limit is investigated by exploiting a connection to the ergodic properties of ows on the unit tangent bundle of a surface of constant negative curvature.

Multidimensional Continued Fractions, Dynamical Renormalization and KAM Theory

AbstractThe disadvantage of ‘traditional’ multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors. Following ideas of Dani, Lagarias and Kleinbock-Margulis we describe a simple algorithm based on the dynamics of flows on the homogeneous space

QUANTUM UNIQUE ERGODICITY FOR PARABOLIC MAPS

SL(d, \mathbb{Z}) \backslash SL(d, \mathbb{R})

Free Path Lengths in Quasicrystals

(the space of lattices of covolume one) that indeed yields best possible approximations to any irrational vector. The algorithm is ideally suited for a number of dynamical applications that involve small divisor problems. As an example, we explicitly construct a renormalization scheme for the linearization of vector fields on tori of arbitrary dimension.

Value distribution of the eigenfunctions and spectral determinants of quantum star graphs

The periodic Lorentz gas describes an ensemble of non-interacting point particles in a periodic array of spherical scatterers. We have recently shown that, in the limit of small scatterer density (Boltzmann–Grad limit), the macroscopic dynamics converges to a stochastic process, whose kinetic transport equation is not the linear Boltzmann equation—in contrast to the Lorentz gas with a disordered scatterer configuration. This paper focuses on the two-dimensional set-up and reports an explicit, elementary formula for the collision kernel of the transport equation.

Unipotent flows on the space of branched covers of Veech surfaces

Abstract. We study the ergodic properties of quantized ergodic maps of the torus. It is known that these satisfy quantum ergodicity: For almost all eigenstates, the expectation values of quantum observables converge to the classical phase-space average with respect to Liouville measure of the corresponding classical observable.¶The possible existence of any exceptional subsequences of eigenstates is an important issue, which until now was unresolved in any example. The absence of exceptional subsequences is referred to as quantum unique ergodicity (QUE). We present the first examples of maps which satisfy QUE: Irrational skew translations of the two-torus, the parabolic analogues of Arnolds cat maps. These maps are classically uniquely ergodic and not mixing. A crucial step is to find a quantization recipe which respects the quantum-classical correspondence principle.¶In addition to proving QUE for these maps, we also give results on the rate of convergence to the phase-space average. We give upper bounds which we show are optimal. We construct special examples of these maps for which the rate of convergence is arbitrarily slow.

Trace formulae for three-dimensional hyperbolic lattices and application to a strongly chaotic tetrahedral billiard

We show that the n-point correlation function for the fractional parts of a random linear form in m variables has a limit distribution with power-like tail. The existence of the limit distribution follows from the mixing property of flows on SL .mC1;R/=SL.mC1;Z/. Moreover, we prove similar limit theorems (i) for the probability to find the fractional part of a random linear form close to zero and (ii) also for related trigonometric sums. For large m, all of the above limit distributions approach the classical distributions for independent uniformly distributed random variables.