Comparison between formulae for the counting functions of the heights t n of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the t n are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian H cl . Many features of H cl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the t n have a similar structure to those of the semiclassical En ; in particular, they display random-matrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the t n can be computed accurately from formulae with quantum analogues. The Riemann-Siegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian H cl = XP. Key words. spectral asymptotics, number theory AMS subject classifications. 11M26, 11M06, 35P20, 35Q40, 41A60, 81Q10, 81Q50 PII. S0036144598347497 1.

The Gutzwiller semiclassical trace formula links the eigenvalues of the Schrodinger operator b H with the closed orbits of the corresponding classical mechanical system, associated with the Hamiltonian H, when the Planck constant is small ("semiclassical r'egime"). Gutzwiller gave a heuristic proof, using the Feynman integral representation for the propagator of b H. Later on mathematicians gave rigorous proofs of this trace formula, under different settings, using the theory of Fourier Integral operators and Lagrangian manifolds. Here we want to show how the use of coherent states (or gaussian beams) allows us to give a simple and direct proof. LPTHE Orsay 97-64 November 1997 1 Laboratoire associ'e au Centre National de la Recherche Scientifique - URA D0063 1 Introduction Our goal in this paper is to give a simple proof of the "semiclassical Gutzwiller trace formula". The pioneering works in quantum physics of Gutzwiller [16] (1971) and BalianBloch [4] [5] (1972-74) showed that th...

Abstract. This paper develops and implements a new algorithm for calculating wave trace invariants of a bounded plane domain around a periodic billiard orbit. The algorithm is based on a new expression for the localized wave trace as a special multiple oscillatory integral over the boundary, and on a Feynman diagrammatic analysis of the stationary phase expansion of the oscillatory integral. The algorithm is particularly effective for Euclidean plane domains possessing a Z2 symmetry which reverses the orientation of a bouncing ball orbit. It is also very effective for domains with dihedral symmetries. For simply connected analytic Euclidean plane domains in either symmetry class, we prove that the domain is determined within the class by either its Dirichlet or Neumann spectrum. This improves and generalizes the previous best inverse result of [Z1, Z2, ISZ] that simply connected analytic plane domains with two symmetries are spectrally determined within that class. 1.

Abstract. This is the first in a series of papers [Z3, Z4, Z5, Z6, Z7] on inverse spectral/resonance problems for analytic plane domains Ω. In this paper, we present a rigorous version of the Balian-Bloch trace formula [BB1, BB2]. It is an asymptotic formula for the trace TrRρ(k + iτ) of the regularized resolvent of the Dirichlet Laplacian of Ω as k → ∞ with τ> 0 held fixed. When the support of ˆρ contains the length Lγ of precisely one periodic reflecting ray γ, then the asymptotic expansion of TrRρ(k + iτ) is essentially the same as the wave trace expansion at γ. The raison d’etre for this approach to wave invariants is that they are explicitly computable. Applications of the trace formula will be given in the subsequent articles in this series. For instance, in [Z4, Z5] we will prove that analytic domains with one symmetry are determined by their Dirichlet spectra. Although we only present details in dimension 2, the Balian-Bloch approach works the same in all dimensions. 1.

Abstract. We obtain asymptotic lower bounds for the spectral function of the Laplacian and for the remainder in local Weyl’s law on manifolds. In the negatively curved case, thermodynamic formalism is applied to improve the estimates. Key ingredients of the proof include the wave equation parametrix, a pretrace formula and the Dirichlet box principle. Our results develop and extend the unpublished thesis of A. Karnaukh [K]. 1. Introduction and

We given an exposition of a proof that a mirror symmetric configuration of two convex analytic obstacles in (R) 2 is determined by its Dirichlet resonance poles. It is the analogue for exterior domains of the proof that a mirror symmetric bounded simply connected analytic plane domain is determined by its Dirichlet eigenvalues. The proof uses ’interior/exterior duality ’ to simplify the argument. 1

Semiclassical approximations often involve the use of stationary phase approximations. This method can be applied when ¯h is small in comparison to relevant actions or action differences in the corresponding classical system. In many situations, however, action differences can be arbitrarily small and then uniform approximations are more appropriate. In the present paper we examine different uniform approximations for describing the spectra of integrable systems and systems with mixed phase space. This is done on the example of two billiard systems, an elliptical billiard and a deformation of it, an oval billiard. We derive a trace formula for the ellipse which involves a uniform approximation for the Maslov phases near the separatrix, and a uniform approximation for tori of periodic orbits close to a bifurcation. We then examine how the trace formula is modified when the ellipse is deformed into an oval. This involves uniform approximations for the break-up of tori and uniform approxi...

We derive uniform approximations for contributions to Gutzwiller's periodic-orbit sum for the spectral density which are valid close to bifurcations of periodic orbits in systems with mixed phase space. There, orbits lie close together and give collective contributions, while the individual contributions of Gutzwiller's type would diverge at the bifurcation. New results for the tangent, the period doubling and the period tripling bifurcation are given. They are obtained by going beyond the local approximation and including higher order terms in the normal form of the action. The uniform approximations obtained are tested on the kicked top and are found to be in excellent agreement with exact quantum results. PACS numbers: 03.20.+i Classical mechanics of discrete systems: general mathematical aspects. 03.65.Sq Semiclassical theories and applications. 05.45.+b Theory and models of chaotic systems. Submitted to Journal of Physics A 1 Introduction Semiclassical approximations in terms ...

We derive semiclassical contributions of periodic orbits from a boundary integral equation for three-dimensional billiard systems. We use an iterative method that keeps track of the composition of the stability matrix and the Maslov index as an orbit is traversed. Results are given for isolated periodic orbits and rotationally invariant families of periodic orbits in axially symmetric billiard systems. A practical method for determining the stability matrix and the Maslov index is described. PACS numbers: 03.65.Sq Semiclassical theories and applications. 05.45.+b Theory and models of chaotic systems. 1 Introduction Billiards are popular models for the study of dynamical systems and their quantum counterparts. They are simpler than general systems with a potential and exhibit a large spectrum of dynamical behaviour ranging from integrability to chaoticity. Until recently, most investigations have concentrated on two-dimensional billiards whose numerical treatment requires much less e...

The vacuum (Casimir) energy of a quantized scalar field in a given geometrical situation is a certain moment of the eigenvalue density of an associated self-adjoint differential operator. For various classes of quantum graphs it has been calculated by several methods: (1) Direct calculation from the explicitly known spectrum is feasible only in simple cases. (2) Analysis of the secular equation determining the spectrum, as in the Kottos–Smilansky derivation of the trace formula, yields a sum over periodic orbits in the graph. (3) Construction of an associated integral kernel by the method of images yields a sum over closed (not necessarily periodic) orbits. We show that for the Kirchhoff and other scale-invariant boundary conditions the sum over nonperiodic orbits in fact makes no contribution to the total energy, whereas for more general (frequency-dependent) vertex scattering matrices it can make a nonvanishing contribution, which, however, is localized near vertices and hence can be “indexed ” a posteriori by truly periodic orbits. For the scale-invariant cases complete calculations have been done by both methods (2) and (3), with identical results. Indeed, applying the image method to the resolvent kernel provides an alternative derivation of the trace formula.