P. Sarnak, Arithmetic Quantum Chaos, Israel Math. Conf. Proceedings, 8, (1995).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....showed, however, that the level statistics on 30 generic (i.e. non arithmetic) surfaces were in nice agreement with the expected random matrix theory prediction in accordance with conjecture 2. This has led Bogomolny, Georgeot, Giannoni, and Schmit [9] Bolte, Steil, and Steiner [10] and Sarnak [11] to introduce the concept of arithmetic quantum chaos. Conjecture 3 (Arithmetic Quantum Chaos) On surfaces of constant negative curvature that are generated by arithmetic fundamental groups, the distribution of the eigenvalues of the quantum Hamiltonian are close to the Poisson distribution. Due ....

....as a check whether we have found all eigenvalues. We then find it necessary to correct one of the terms in [39] numerically. Finally, we regard the spectral fluctuations and find that the nearest neighbor spacing distribution closely resembles that of a Poisson random process as predicted by [9, 10, 11] and previously observed by [32] In the first step we consider the level counting function N(r) # i r i r and split it into two parts N(r) N fluc (r) Here N is a smooth function describing the average increase in the number of levels and N fluc describes the fluctuations ....

P. Sarnak. Arithmetic quantum chaos. Israel Math. Conf. Proc., 8:183--236, 1995.

....The fact that the semi classical measure of any sequence of eigenfunctions (with increasing eigenvalues) is invariant by the action of the hamiltonian flow is a consequence of the propagation results (Lemma 4. 1) Apart from the beautifull, more precise but particular results on arithmetic surfaces [34], 31] this result seems to be the first one showing that there exist some invariant measures which are not the semi classical measures of any sequence of eigenfunctions. We are going to deduce Theorem 3 # from the following result [4] which is related to some earlier control results of Haraux ....

P. Sarnak. Arithmetic quantum chaos. In The Schur lectures (1992.

....which is still mysterious to mathematicians and physicists alike is if the states of this system (that is, solutions of the equation above) can concentrate on the highly ustable closed orbits of the classical billiard. Quantum unique ergodicity states that there is no such concentration see [8], 9] 6] and references given there. In the arithmetic case, that is for billiards given by arithmetic surfaces where the motion is given by the geodesic flow, spectacular advances have been recently achieved by Bourgain, Lindenstrauss [10] and Sarnak, while for the popular quantization of the ....

P. Sarnak. Arithmetic quantum chaos. In The Schur lectures (1992.

....this should not be possible. In [CdV] for example, it is conjectured that such sequences should not exist on constant negative curvature surfaces. Partial results in this direction have been obtained using number theoretic methods for certain arithmetic hyperbolic surfaces [RS] LS] S3] see [S1], S2] for a review) In this paper we analyze the above problem for a simpler class of models that has attracted much attention, namely the quantized ergodic automorphisms of the 2d torus. We prove here that for these models such sequences do not exist (Theorem 1.1) We also obtain a stronger ....

Sarnak P., Arithmetic quantum chaos, Israel Math. Conf. Proc. 8, 183-236 (1995).

....should not exist [DEGI] KR1] KR2] The existence or absence of some (possibly weaker) form of scarring is in general an interesting question in quantum chaos, that can be asked also in the context of billiards, for example. For more information on the question of scarring, you may consult [S1] and references therein. 14.2. Eigenvalue behaviour. The attentive reader should be disconcerted to see the end of these notes approach without me returning to the problems of eigenvalue statistics and the Berry Tabor and Bohigas Giannoni Schmit conjectures with which I started them. Did I not ....

....surfaces, nor trace formulas have even been mentioned and the links between quantum chaos and random matrix theory or number theory have been completely ignored. To explore the subject further I recommend you the literature already cited, as well as some of the many review articles available: B][S1][S2] GMW] I hope these notes have at least furnished an adequate introduction to those more advanced treatments of the subject. Appendix A. Torus translations In the notations of section 8, consider T . For notational simplicity, I choose a i = e i ; i = 1 : n, where fe i g i=1: n ....

Sarnak P., Arithmetic quantum chaos, Israel Math. Conf. Proc. 8, 183-236 (1995).

....and Sarnak [L S3] It is known (see Section 7) that on an n dimensional manifold, jj jj 1 = O( n 1) 4 ) for = 0; jj jj 2 = 1. This bound was improved in [I S1, Sar5] for arithmetic hyperbolic surfaces, and in [Ko1] for compact arithmetic hyperbolic 3 manifolds. It was conjectured in [Sar2] that on negatively curved surfaces jj jj 1 = O( for any 3 The existence of L 2 eigenfunctions of for non compact hyperbolic surfaces of nite area depends very strongly on whether surface is arithmetic or not; it is conjectured that L 2 eigenfunctions generally do not exist for ....

P. Sarnak. Arithmetic quantum chaos. Israel Math. Conf. Proc. 8 (1995), 183-236.

....short range correlations are numerically found to be in accordance with random matrix theory (RMT) 4, 5, 6, 7] whereas long range correlations experience modifications leading again to saturation effects. This general behaviour of chaotic systems is violated by the class of arithmetical systems [8, 9, 10, 11, 12, 13], where, e.g. the level spacing distribution and the two point statistics nearly behave as for classically integrable systems. In most numerical studies concerning the statistical properties of quantal levels of chaotic systems, the nearest neighbour level spacing distribution P (0; L) the ....

....to short range correlations because of, e.g. the saturation of the number variance and rigidity, contrasting the expected logarithmic increase in RMT. Furthermore, even for short range correlations deviations were found for a class of strongly chaotic systems, the so called arithmetical systems [9, 10, 11, 13]. Therefore in [14, 15] a different statistical measure as signature of quantum chaos is proposed which is given by the probability distribution of the suitably normalized fluctuations of N fluc (x) Figure 3: N fluc (x) for the integrable circular billiard. In order to define this statistical ....

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P. Sarnak, Arithmetic quantum chaos, in: Israel Math. Conference Proceedings 8(1995) 183.

....is based on talks given at the Workshop on Number Theoretic Aspects of Ergodic Theory and Quantum Chaos, St. Andreasberg Harz, June 17 21, 1996, and at the IMA Workshop on Emerging Applications of Number Theory, Minneapolis, July 15 26, 1996. 1 2 GUNTHER STEIL quantum chaos was given by Sarnak [24], see also [4,7] In this article we study three dimensional analogues of the situation described. We want to gain some insight into the discrete spectra of the Laplacian for Bianchi groups PSL(2; O) where O is the ring of integers of an imaginary quadratic number field. These are arithmetic ....

P. Sarnak, Arithmetic quantum chaos, Israel Math. Conf. Proc., 8 (1995), pp. 183--236.

....be) or have many possible generalizations (e.g. the concept of entropy) and it is unclear which should be used to produce results analogous to the classical setting. Some interesting attempts to address such questions have been done (notably all the work under the label of Quantum Chaos, see [72] for some rigorous results and [36] 30] for an overview of the field) Yet, the situation is still vastly unsatisfactory. This state of affair reflects on the fact that, although I have made an effort to present the classical theory in a manner as close as possible to the quantum setting, it is ....

Sarnak P. Arithmetic quantum chaos, Israel Math. Conf. Proc. 8 (1995), 183--236.

....A group G is a Coxeter group, if it is a group of isometries of the (spherical) hyperbolic or Euclidean plane generated by a nite set of re ections, through (great circles or) hyperplanes. The set S of generating re ections is called the set of standard generators of the Coxeter group. 7 See [30] and the references therein for some numerical evidence pertaining to (at least the rst of) these questions. 14 We will restrict our discussion to the case when the quotient space has nite volume (in the case when the fundamental domain is a polygon) Let (G) be the orbifold Euler ....

P. Sarnak. Arithmetic quantum chaos. Israel Math. Conf. Proc., 8:183-236, 1995.

....in the ergodic properties of non relativistic quantum systems. For example, on the one hand a large literature has been devoted to the so called quantum chaos, that is to the study of the spectrum of the Hamiltonians whose classical associated motion enjoys strong ergodic properties (see [12, 16, 28] for an overview) On the other hand, there has been renewed interested in the study of the convergence to equilibrium in quantum systems (i.e. in the study of strong statistical properties directly in the quantum setting) 11, 13, 20, 24, 25, 33] Of course, in order for such ergodic properties ....

Sarnak P. Arithmetic quantum chaos, Israel Math. Conf. Proc. 8 (1995), 183-- 236.

.... 2 p = p(d) 4, then by Hausdorff Young theorem 0 X 2Z d jb j q 1 A 1=q jj jj 2 2 uniformly on T d for q = q(d) p(d) p(d) Gamma 2) 4 The question of the rate of growth of the ratio in (4) with increasing eigenvalue is one of the basic questions in quantum chaos, cf. [Sar]. 5 We can identify the eigenfunctions on T d with the eigenfunctions on T d 2 all of whose frequencies lie in the subspace fx 2 R d 2 jx d 1 = x d 2 = 0g of R d 2 . ON QUANTUM LIMITS ON FLAT TORI 83 Accordingly, we define a number C(d; q) 1 by (6) C(d; q) sup Delta =0 (jj b f ....

P. Sarnak, Arithmetic Quantum Chaos, Blyth Lectures, Toronto (1993).

....their perturbations) as well as the aforementioned discontinuous maps. Note that among those only the geodesic flows and the (smoothly perturbed) toral automorphisms are smooth. Their behaviour is often determined by special number theoretic properties and therefore not expected to be generic [HB, Ke1, Ke2, S]. This has been a major motivation for analyzing systems with singularities. Much attention has been paid to chaotic maps on the torus, even though they seem rather unrealistic as models for physical systems despite a recent attempt to realize the quantum Baker map as a physically realistic system ....

.... 2I j e k ; j 2 : Here the e k , k = 0; N Gamma 1, are the usual position eigenvectors (see Section 2 for the precise definition) Theorem 2 has now the following main corollary, which is the usual statement of the equidistribution of the eigenfunctions in the position variables [CdV, S, Z1]. Corollary 3 Under the assumptions of Theorem 2: j N (I) X k N 2I j e k ; j N j 2 Gamma jIj; when N 1 The paper is organized as follows. In section 2 we briefly recall the basic elements of quantization on the torus. In section 3 we prove an Egorov estimate for quantized ....

P. Sarnak, Arithmetic quantum chaos, Tel Aviv Lectures (1993).

.... average of the sub principal symbol oe sub (A ) does not vanish, then a lower bound is given by [3] j ( A Psi n ; Psi n ) Gamma oe A j = Omega E Gamma 1 2 n : 8) Addressing the question of quantum ergodicity in individual eigenstates, a conjecture is put forward by Sarnak [14] which states that for hyperbolic surfaces in which the curvature is negative ( A Psi n ; Psi n ) Gamma oe A = O E Gamma 1 4 n for all 0 : 9) The main topic in this paper deals with the rate of the decay of S 1 (E; A) An extended analysis including higher S k (E; A) k 1, ....

P. Sarnak, "Arithmetic quantum chaos", in: Israel Math. Conference Proceedings 8(1995) 183.

....of the differences i 1 Gamma i . The question is whether the histogram looks like the graph of e Gammax . Question 3 is a finite Euclidean analogue of questions considered in mathematical physics [Courant and Hilbert 1961, p. 302] See also [Gutzwiller 1990; Hejhal and Rackner 1992; Sarnak 1995] In the case of eigenfunctions of Delta = 2 = x 2 2 = y 2 that are zero on the boundary of a domain D in the real plane R 2 , we are asking for the points of a vibrating drum that reach a given 15 August 1996 at 18:54 Terras: Survey of Spectra of Laplacians on Finite ....

P. Sarnak, "Arithmetic quantum chaos", pp. 183--236 in The Schur Lectures (Tel Aviv, 1992), Israel Math. Soc. Conf. Proc. 8, Bar-Ilan University, Ramat-Gan, 1995.

....definition (1.6) reproduces the classical one at the classical limit if hu n ; Au n i tends to the phase average of the symbol of A, as verified in many instances (see e.g. Sc, CdV, HMR, Ze1, Ze2, DEGI] in which H is the quantization of a Hamiltonian generating an ergodic flow. Some authors ([Sa, Ze2] assume this limiting property as the very definition of quantum ergodicity. Ergodic Properties of Quantum Harmonic Crystals 29 ....

P.Sarnak, Arithmetic Quantum Chaos, Tel Aviv Lectures, (1994), 181182

....and Tabor [2] proposed that at length scales which give unit mean level spacing between eigenvalues, the distribution of their differences should be uniform for quantum systems which are classically completely integrable. That conjecture has been verified numerically [4] in some cases see also [11] for more references. The only cases in which mathematical results have been obtained are those of Zoll surfaces (spheres with metrics for which all of the geodesics are closed) and of flat tori. For Zoll surfaces, Uribe and the first author [15] showed the existence of a suitably defined pair ....

Peter Sarnak, Arithmetic quantum chaos. Schur lectures, Israel Math. Conf.Proc. 8(1995).

....of various invariants of the spectrum f j g and eigenfunctions fOE j g in the limit j 1, under the condition that G t acts ergodically with respect to the normalized Liouville measure d on S M . For some of the many heuristic and numerical results we refer to the recent survey of Sarnak [Sa]. From the C algebra point of view, a quantum dynamical system is a C dynamical system (A; G; ff) where A is a C algebra, and ff : G Aut(A) is a representation of G by automorphisms of A. We will always assume A is unital and separable, that G is amenable and that the system is ....

P. Sarnak, Arithmetic Quantum Chaos (to appear).

....connection between the distribution of closed orbits in phase space and the localization properties of the quantum matrix elements. In the analogous case of hyperbolic surfaces, the relation of individual Wigner functions of eigenfunctions to periodic orbits is at best very unclear (see e.g.[Sa]) We give now some easy consequences of Theorem 4.1. 40 MIRKO DEGLI ESPOSTI, SANDRO GRAFFI, STEFANO ISOLA Corollary 4.1. Let Phi be an eigenvector in H and W = fW OE (k) g k1 the corresponding Wigner function. Then W OE (k) Gamma 1 for k 1 (4:20) in the weak topology of A(T 2 ) ....

P.Sarnak, Arithmetic Quantum Chaos, Tel Aviv Lectures 1993 (to appear). 46 MIRKO DEGLI ESPOSTI, SANDRO GRAFFI, STEFANO ISOLA

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P. Sarnak, Arithmetic Quantum Chaos, Israel Math. Conf. Proceedings, 8, (1995).

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P. Sarnak, Arithmetic Quantum Chaos, The R.A. Blyth Lecture, University of Toronto (1993).

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P. Sarnak, Arithmetic Quantum Chaos, Israel Math. Conf. Proceedings, 8, (1995).

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P. Sarnak, Arithmetic quantum chaos, Israel Math. Soc. Conf. Proc., Schur Lectures, Bar-Ilan Univ., Ramat-Gan, Israel, 1995.

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P. Sarnak, "Arithmetic quantum chaos", Tel Aviv Lectures (1993)

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